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

<span class="kw">let </span><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/>  for <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">N</font> being   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">G</font><br/>  for <font color="Olive" title="b4">MN</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">N</font>  st <font color="Olive" title="b4">MN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b2">M</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">N</font> holds <br/><font color="Olive" title="b3">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b4">MN</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b3">MN</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">N</font>  st <font color="Olive" title="b3">MN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b1">M</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">N</font> holds <br/><font color="Olive" title="b2">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">MN</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">N</font> be   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">MN</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">N</font>  st <font color="Olive" title="b1">MN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c3">M</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">N</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b1">MN</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">M9</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">M</font> #) as   <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="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E20"><span class="lab"><font color="Green" title="E19">Lm7</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">N9</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">N</font> #) as   <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="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E20"><span class="lab"><font color="Green" title="E19">Lm7</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">MN</font> be   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">MN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c3">M</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">N</font> implies <font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:118"/><span class="lab"><font color="Green" title="E96">A1</font></span>: 
<font color="Maroon" title="c7">MN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">M</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">N</font> or <font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">MN99</font> = (<font color="Maroon" title="c6">N9</font>,<font color="Maroon" title="c5">M9</font>) <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a>  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="c6">N9</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">MN9</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">MN</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c7">MN</font> #) as   <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="c4">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E20"><span class="lab"><font color="Green" title="E19">Lm7</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:118"/>
<font color="Maroon" title="c3">M</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E7:118"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/>
<a NAME="E8:118"/><span class="kw">then </span>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font> &amp;  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">N</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/>
<a NAME="E9:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A2</font></span>: 
<font color="Maroon" title="c5">M9</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c6">N9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/>
<a NAME="E10:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A3</font></span>: 
(<font color="Maroon" title="c6">N9</font>,<font color="Maroon" title="c5">M9</font>) <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">MN9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="ref" href="group_6.html#D1" title="GROUP_6:def.1">GROUP_6:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">K</font> = <font color="Maroon" title="c6">N9</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c6">N9</font>,<font color="Maroon" title="c5">M9</font>) <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a></span>)</span> as   <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="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c5">M9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:118"><span class="lab"><font color="Green" title="E97">A2</font></span></a>, <a class="ref" href="group_6.html#T29" title="GROUP_6:th.29">GROUP_6:29</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E99">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c8">MN99</font> ) &amp; ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c8">MN99</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c8">MN99</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:118_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c8">MN99</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E3:118_1_1"/><span class="lab"><font color="Green" title="E100">A5</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">and </span><br/><a NAME="E4:118_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">MN9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T13" title="GROUP_6:th.13">GROUP_6:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a9</font> = <font color="Maroon" title="c12">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">A</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">A9</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">a9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">N9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span>  implies <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A9</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span>  ) &amp; ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A9</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span>  implies <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span>  ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A9</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span>  implies <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:118_1_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A9</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">g</font>, <font color="Maroon" title="c18">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E3:118_1_1_1_1"/><span class="lab"><font color="Green" title="E101">A6</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c18">h</font>
 <span class="kw">and </span><br/><a NAME="E4:118_1_1_1_1"/><span class="lab"><font color="Green" title="E102">A7</font></span>: 
( <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c18">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">h9</font> = <font color="Maroon" title="c18">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">g9</font> = <font color="Maroon" title="c17">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> ;<br/><a NAME="E7:118_1_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_1_1_1_1"><span class="lab"><font color="Green" title="E101">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:118_1_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A9</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:118"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E4:118_1_1_1_1"><span class="lab"><font color="Green" title="E102">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:118_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A9</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">g</font>, <font color="Maroon" title="c18">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font><span class="kw"> such that </span><br/><a NAME="E4:118_1_1_1"/><span class="lab"><font color="Green" title="E101">A8</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c18">h</font>
 <span class="kw">and </span><br/><a NAME="E5:118_1_1_1"/><span class="lab"><font color="Green" title="E102">A9</font></span>: 
( <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A9</font> &amp; <font color="Maroon" title="c18">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">h9</font> = <font color="Maroon" title="c18">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">g9</font> = <font color="Maroon" title="c17">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><a NAME="E8:118_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_1_1_1"><span class="lab"><font color="Green" title="E101">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:118_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:118"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E5:118_1_1_1"><span class="lab"><font color="Green" title="E102">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:118_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c8">MN99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_1_1"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><span class="kw">hence </span><a NAME="E10:118_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c8">MN99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T14" title="GROUP_6:th.14">GROUP_6:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:118_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c8">MN99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font><span class="kw"> such that </span><br/><a NAME="E4:118_1"/><span class="lab"><font color="Green" title="E100">A10</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c8">MN99</font>
 <span class="kw">and </span><br/><a NAME="E5:118_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">MN99</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c12">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T13" title="GROUP_6:th.13">GROUP_6:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">a9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">A9</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">N9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">A</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span>  implies <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span>  ) &amp; ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span>  implies <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span>  ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span>  implies <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:118_1_2_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">g</font>, <font color="Maroon" title="c18">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E3:118_1_2_1"/><span class="lab"><font color="Green" title="E101">A11</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c18">h</font>
 <span class="kw">and </span><br/><a NAME="E4:118_1_2_1"/><span class="lab"><font color="Green" title="E102">A12</font></span>: 
( <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Maroon" title="c18">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">h9</font> = <font color="Maroon" title="c18">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">g9</font> = <font color="Maroon" title="c17">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> ;<br/><a NAME="E7:118_1_2_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_1_2_1"><span class="lab"><font color="Green" title="E101">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:118_1_2_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:118"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E4:118_1_2_1"><span class="lab"><font color="Green" title="E102">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:118_1_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">g</font>, <font color="Maroon" title="c18">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font><span class="kw"> such that </span><br/><a NAME="E4:118_1_2"/><span class="lab"><font color="Green" title="E101">A13</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c18">h</font>
 <span class="kw">and </span><br/><a NAME="E5:118_1_2"/><span class="lab"><font color="Green" title="E102">A14</font></span>: 
( <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> &amp; <font color="Maroon" title="c18">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c8">MN99</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">h9</font> = <font color="Maroon" title="c18">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">g9</font> = <font color="Maroon" title="c17">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><a NAME="E8:118_1_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_1_2"><span class="lab"><font color="Green" title="E101">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:118_1_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h99</font></span>)</span> where <font color="Olive" title="b1">g99</font>, <font color="Olive" title="b2">h99</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : ( <font color="Olive" title="b1">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Olive" title="b2">h99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:118"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E5:118_1_2"><span class="lab"><font color="Green" title="E102">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:118_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_1"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:118_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T14" title="GROUP_6:th.14">GROUP_6:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="lab"><font color="Green" title="E100">A15</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">K</font> = 
 <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>
;<br/>
<div><span class="lab"><font color="Green" title="E101">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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 <font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> #) implies <font color="Maroon" title="c11">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:118_3"/><span class="lab"><font color="Green" title="E102">A17</font></span>: 
<font color="Maroon" title="c11">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">a</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="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c12">a</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a9</font> = <font color="Maroon" title="c12">a</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="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c5">M9</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c11">H</font></span>)</span> implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c11">H</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c12">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:118_3_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c11">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c11">H</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c12">a</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">b</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="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:118_3_1_1"/><span class="lab"><font color="Green" title="E103">A18</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">and </span><br/><a NAME="E4:118_3_1_1"/><span class="lab"><font color="Green" title="E104">A19</font></span>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c11">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">b9</font> = <font color="Maroon" title="c15">b</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="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c5">M9</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><a NAME="E6:118_3_1_1"/><span class="lab"><font color="Green" title="E105">A20</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_3_1_1"><span class="lab"><font color="Green" title="E103">A18</font></span></a>, <a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x9</font> = <font color="Maroon" title="c14">x</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="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c5">M9</font></span>)</span> ;<br/><a NAME="E8:118_3_1_1"/>
( <font color="Maroon" title="c13">a9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c10">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">K</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c13">a9</font> &amp; <font color="Maroon" title="c17">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">a9</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c10">K</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:118"><span class="lab"><font color="Green" title="E100">A15</font></span></a>, <a class="txt" href="#E1:118_3"><span class="lab"><font color="Green" title="E102">A17</font></span></a>, <a class="txt" href="#E4:118_3_1_1"><span class="lab"><font color="Green" title="E104">A19</font></span></a>, <a class="txt" href="#E6:118_3_1_1"><span class="lab"><font color="Green" title="E105">A20</font></span></a>, <a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a>, <a class="ref" href="group_3.html#T118" title="GROUP_3:th.118">GROUP_3:118</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">c9</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="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c5">M9</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:118_3_1_1"/><span class="lab"><font color="Green" title="E106">A21</font></span>: 
<font color="Maroon" title="c17">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">c9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a9</font>
 <span class="kw">and </span><br/><a NAME="E11:118_3_1_1"/><span class="lab"><font color="Green" title="E107">A22</font></span>: 
<font color="Maroon" title="c18">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T28" title="GROUP_2:th.28">GROUP_2:28</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">c</font> = <font color="Maroon" title="c18">c9</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="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><a NAME="E13:118_3_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:118_3_1_1"><span class="lab"><font color="Green" title="E106">A21</font></span></a>, <a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></span>;<br/><span class="kw">hence </span><a NAME="E14:118_3_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c11">H</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:118"><span class="lab"><font color="Green" title="E100">A15</font></span></a>, <a class="txt" href="#E1:118_3"><span class="lab"><font color="Green" title="E102">A17</font></span></a>, <a class="txt" href="#E11:118_3_1_1"><span class="lab"><font color="Green" title="E107">A22</font></span></a>, <a class="ref" href="group_2.html#T28" title="GROUP_2:th.28">GROUP_2:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:118_3_1"/>
<font color="Maroon" title="c12">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:118_3"/>
<font color="Maroon" title="c11">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T118" title="GROUP_3:th.118">GROUP_3:118</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:118"/><span class="lab"><font color="Green" title="E102">A23</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c5">M9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/>
<a NAME="E16:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A24</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:118"><span class="lab"><font color="Green" title="E100">A15</font></span></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E104">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_4_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="E1:118_4_1"/>
(  not <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> or <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:118_4_1_1"/><span class="lab"><font color="Green" title="E105">A26</font></span>: 
 not <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span></span><br/><div class="add"><a NAME="E2:118_4_1_1"/><span class="lab"><font color="Green" title="E106">A27</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:118"><span class="lab"><font color="Green" title="E102">A23</font></span></a>, <a class="txt" href="#E13:118"><span class="lab"><font color="Green" title="E100">A15</font></span></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/><div><span class="lab"><font color="Green" title="E107">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_4_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:118_4_1_1_1"/><span class="lab"><font color="Green" title="E108">A29</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span></span><br/><a NAME="E2:118_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A30</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></span>;<br/><a NAME="E3:118_4_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_1_1"><span class="lab"><font color="Green" title="E108">A29</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></span>;<br/><a NAME="E4:118_4_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118_4_1_1"><span class="lab"><font color="Green" title="E106">A27</font></span></a>, <a class="txt" href="#E2:118_4_1_1_1"><span class="lab"><font color="Green" title="E109">A30</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E5:118_4_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118_4_1_1_1"><span class="lab"><font color="Green" title="E109">A30</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E108">A31</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_4_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:118_4_1_1_2"/><span class="lab"><font color="Green" title="E109">A32</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span></span><br/><a NAME="E2:118_4_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/><a NAME="E3:118_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A33</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></span>;<br/><a NAME="E4:118_4_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_1_2"><span class="lab"><font color="Green" title="E109">A32</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E5:118_4_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_4_1_1_2"><span class="lab"><font color="Green" title="E110">A33</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font> = 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_1"><span class="lab"><font color="Green" title="E105">A26</font></span></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_4_1_1"><span class="lab"><font color="Green" title="E107">A28</font></span></a>, <a class="txt" href="#E4:118_4_1_1"><span class="lab"><font color="Green" title="E108">A31</font></span></a>, <a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_1"><span class="lab"><font color="Green" title="E105">A26</font></span></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="118_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:118_4_1_2"/><span class="lab"><font color="Green" title="E105">A34</font></span>: 
<font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span></span><br/><div class="add"><a NAME="E2:118_4_1_2"/><span class="kw">then </span>
 the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:118_4_1_2"/><span class="lab"><font color="Green" title="E106">A35</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font>
 <span class="kw">and </span><br/><a NAME="E5:118_4_1_2"/><span class="lab"><font color="Green" title="E107">A36</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:118_4_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><a NAME="E7:118_4_1_2"/><span class="lab"><font color="Green" title="E108">A37</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">M</font> :  ex <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">a</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_2"><span class="lab"><font color="Green" title="E105">A34</font></span></a>, <a class="txt" href="#E4:118_4_1_2"><span class="lab"><font color="Green" title="E106">A35</font></span></a>, <a class="ref" href="group_9.html#D16" target="_self" title="GROUP_9:def.16">Def16</a></span>;<br/><a NAME="E8:118_4_1_2"/>
 the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_2"><span class="lab"><font color="Green" title="E105">A34</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E10:118_4_1_2"/><span class="lab"><font color="Green" title="E109">A38</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font>
 <span class="kw">and </span><br/><a NAME="E11:118_4_1_2"/><span class="lab"><font color="Green" title="E110">A39</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:118_4_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><a NAME="E13:118_4_1_2"/><span class="lab"><font color="Green" title="E111">A40</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:118"><span class="lab"><font color="Green" title="E103">A24</font></span></a>, <a class="txt" href="#E5:118_4_1_2"><span class="lab"><font color="Green" title="E107">A36</font></span></a>, <a class="txt" href="#E11:118_4_1_2"><span class="lab"><font color="Green" title="E110">A39</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E14:118_4_1_2"/><span class="lab"><font color="Green" title="E112">A41</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c7">MN</font> :  ex <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">a</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_2"><span class="lab"><font color="Green" title="E105">A34</font></span></a>, <a class="txt" href="#E10:118_4_1_2"><span class="lab"><font color="Green" title="E109">A38</font></span></a>, <a class="ref" href="group_9.html#D16" target="_self" title="GROUP_9:def.16">Def16</a></span>;<br/><div><span class="lab"><font color="Green" title="E113">A42</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_4_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font> implies <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:118_4_1_2_1"/><span class="lab"><font color="Green" title="E114">A43</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span><br/><a NAME="E2:118_4_1_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">A2</font>, <font color="Maroon" title="c16">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c7">MN</font><span class="kw"> such that </span><br/><a NAME="E4:118_4_1_2_1"/><span class="lab"><font color="Green" title="E115">A44</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">A2</font>,<font color="Maroon" title="c16">B2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:118_4_1_2_1"/><span class="lab"><font color="Green" title="E116">A45</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A2</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B2</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:118_4_1_2"><span class="lab"><font color="Green" title="E112">A41</font></span></a></span>;<br/><a NAME="E6:118_4_1_2_1"/><span class="lab"><font color="Green" title="E117">A46</font></span>: 
<font color="Maroon" title="c15">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_4_1_2_1"><span class="lab"><font color="Green" title="E115">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E7:118_4_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:118"><span class="lab"><font color="Green" title="E103">A24</font></span></a>, <a class="txt" href="#E5:118_4_1_2"><span class="lab"><font color="Green" title="E107">A36</font></span></a>, <a class="txt" href="#E11:118_4_1_2"><span class="lab"><font color="Green" title="E110">A39</font></span></a>, <a class="txt" href="#E1:118_4_1_2_1"><span class="lab"><font color="Green" title="E114">A43</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">A1</font>, <font color="Maroon" title="c18">B1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">M</font><span class="kw"> such that </span><br/><a NAME="E9:118_4_1_2_1"/><span class="lab"><font color="Green" title="E118">A47</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c18">B1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:118_4_1_2_1"/><span class="lab"><font color="Green" title="E119">A48</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118_4_1_2"><span class="lab"><font color="Green" title="E108">A37</font></span></a></span>;<br/><a NAME="E11:118_4_1_2_1"/><span class="lab"><font color="Green" title="E120">A49</font></span>: 
<font color="Maroon" title="c17">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:118_4_1_2_1"><span class="lab"><font color="Green" title="E118">A47</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">A29</font> = <font color="Maroon" title="c15">A2</font>, <font color="Maroon" title="c20">B29</font> = <font color="Maroon" title="c16">B2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">A19</font> = <font color="Maroon" title="c17">A1</font>, <font color="Maroon" title="c22">B19</font> = <font color="Maroon" title="c18">B1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c5">M9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">fo</font> = <font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font>;<br/><a NAME="E14:118_4_1_2_1"/>
<font color="Maroon" title="c4">N</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/><a NAME="E15:118_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A50</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">a2</font>, <font color="Maroon" title="c25">b2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E17:118_4_1_2_1"/><span class="lab"><font color="Green" title="E122">A51</font></span>: 
<font color="Maroon" title="c24">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A2</font>
 <span class="kw">and </span><br/><a NAME="E18:118_4_1_2_1"/><span class="lab"><font color="Green" title="E123">A52</font></span>: 
<font color="Maroon" title="c25">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B2</font>
 <span class="kw">and </span><br/><a NAME="E19:118_4_1_2_1"/><span class="lab"><font color="Green" title="E124">A53</font></span>: 
<font color="Maroon" title="c25">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:118_4_1_2_1"><span class="lab"><font color="Green" title="E116">A45</font></span></a></span>;<br/><a NAME="E20:118_4_1_2_1"/><span class="lab"><font color="Green" title="E125">A54</font></span>: 
<font color="Maroon" title="c20">B29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">b2</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:118_4_1_2_1"><span class="lab"><font color="Green" title="E123">A52</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><a NAME="E21:118_4_1_2_1"/>
( <font color="Maroon" title="c24">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c25">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font> )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">a29</font> = <font color="Maroon" title="c24">a2</font>, <font color="Maroon" title="c27">b29</font> = <font color="Maroon" title="c25">b2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:118_4_1_2_1"><span class="lab"><font color="Green" title="E121">A50</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c28">a1</font>, <font color="Maroon" title="c29">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E24:118_4_1_2_1"/><span class="lab"><font color="Green" title="E126">A55</font></span>: 
<font color="Maroon" title="c28">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A1</font>
 <span class="kw">and </span><br/><a NAME="E25:118_4_1_2_1"/><span class="lab"><font color="Green" title="E127">A56</font></span>: 
<font color="Maroon" title="c29">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B1</font>
 <span class="kw">and </span><br/><a NAME="E26:118_4_1_2_1"/><span class="lab"><font color="Green" title="E128">A57</font></span>: 
<font color="Maroon" title="c29">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:118_4_1_2_1"><span class="lab"><font color="Green" title="E119">A48</font></span></a></span>;<br/><a NAME="E27:118_4_1_2_1"/><span class="lab"><font color="Green" title="E129">A58</font></span>: 
<font color="Maroon" title="c21">A19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a1</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:118_4_1_2_1"><span class="lab"><font color="Green" title="E126">A55</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_4_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span> implies <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">b29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span> ) &amp; ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">b29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span> implies <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">b29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span> implies <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:118_4_1_2_1_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">b29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E3:118_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E130">A59</font></span>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c31">h</font>
 <span class="kw">and </span><br/><a NAME="E4:118_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E131">A60</font></span>: 
<font color="Maroon" title="c31">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>;<br/><a NAME="E5:118_4_1_2_1_1_1"/>
<font color="Maroon" title="c31">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">h9</font> = <font color="Maroon" title="c31">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:118_4_1_2_1"><span class="lab"><font color="Green" title="E121">A50</font></span></a></span>;<br/><a NAME="E7:118_4_1_2_1_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">b29</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c32">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E130">A59</font></span></a>, <a class="ref" href="group_9.html#T3" target="_self" title="GROUP_9:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:118_4_1_2_1_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">b29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E4:118_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E131">A60</font></span></a>, <a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:118_4_1_2_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">b29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E4:118_4_1_2_1_1"/><span class="lab"><font color="Green" title="E130">A61</font></span>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">b29</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c31">h</font>
 <span class="kw">and </span><br/><a NAME="E5:118_4_1_2_1_1"/><span class="lab"><font color="Green" title="E131">A62</font></span>: 
<font color="Maroon" title="c31">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>;<br/><a NAME="E6:118_4_1_2_1_1"/>
<font color="Maroon" title="c31">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E5:118_4_1_2_1_1"><span class="lab"><font color="Green" title="E131">A62</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">h9</font> = <font color="Maroon" title="c31">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><a NAME="E8:118_4_1_2_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c32">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_4_1_2_1_1"><span class="lab"><font color="Green" title="E130">A61</font></span></a>, <a class="ref" href="group_9.html#T3" target="_self" title="GROUP_9:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:118_4_1_2_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E5:118_4_1_2_1_1"><span class="lab"><font color="Green" title="E131">A62</font></span></a>, <a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E29:118_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A63</font></span>: 
<font color="Maroon" title="c27">b29</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">M9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">b2</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><a NAME="E30:118_4_1_2_1"/><span class="lab"><font color="Green" title="E131">A64</font></span>: 
<font color="Maroon" title="c16">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_4_1_2_1"><span class="lab"><font color="Green" title="E115">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E31:118_4_1_2_1"/><span class="lab"><font color="Green" title="E132">A65</font></span>: 
<font color="Maroon" title="c18">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:118_4_1_2_1"><span class="lab"><font color="Green" title="E118">A47</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_4_1_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">a2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span> implies <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">a29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span> ) &amp; ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">a29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span> implies <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">a2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">a29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span> implies <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">a2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:118_4_1_2_1_2_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">a2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">a29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E3:118_4_1_2_1_2_1"/><span class="lab"><font color="Green" title="E133">A66</font></span>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c31">h</font>
 <span class="kw">and </span><br/><a NAME="E4:118_4_1_2_1_2_1"/><span class="lab"><font color="Green" title="E134">A67</font></span>: 
<font color="Maroon" title="c31">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>;<br/><a NAME="E5:118_4_1_2_1_2_1"/>
<font color="Maroon" title="c31">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">h9</font> = <font color="Maroon" title="c31">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:118_4_1_2_1"><span class="lab"><font color="Green" title="E121">A50</font></span></a></span>;<br/><a NAME="E7:118_4_1_2_1_2_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a29</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c32">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_4_1_2_1_2_1"><span class="lab"><font color="Green" title="E133">A66</font></span></a>, <a class="ref" href="group_9.html#T3" target="_self" title="GROUP_9:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:118_4_1_2_1_2_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">a29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E4:118_4_1_2_1_2_1"><span class="lab"><font color="Green" title="E134">A67</font></span></a>, <a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:118_4_1_2_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">a29</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">a2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E4:118_4_1_2_1_2"/><span class="lab"><font color="Green" title="E133">A68</font></span>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a29</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c31">h</font>
 <span class="kw">and </span><br/><a NAME="E5:118_4_1_2_1_2"/><span class="lab"><font color="Green" title="E134">A69</font></span>: 
<font color="Maroon" title="c31">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>;<br/><a NAME="E6:118_4_1_2_1_2"/>
<font color="Maroon" title="c31">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E5:118_4_1_2_1_2"><span class="lab"><font color="Green" title="E134">A69</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">h9</font> = <font color="Maroon" title="c31">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><a NAME="E8:118_4_1_2_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c32">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_4_1_2_1_2"><span class="lab"><font color="Green" title="E133">A68</font></span></a>, <a class="ref" href="group_9.html#T3" target="_self" title="GROUP_9:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:118_4_1_2_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">a2</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c9">MN9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E5:118_4_1_2_1_2"><span class="lab"><font color="Green" title="E134">A69</font></span></a>, <a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E33:118_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A70</font></span>: 
<font color="Maroon" title="c24">a2</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c9">MN9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a29</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><a NAME="E34:118_4_1_2_1"/>
<font color="Maroon" title="c19">A29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">a2</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c9">MN9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:118_4_1_2_1"><span class="lab"><font color="Green" title="E122">A51</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><a NAME="E35:118_4_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">a1</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="c26">a29</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:118_4_1_2_1"><span class="lab"><font color="Green" title="E120">A49</font></span></a>, <a class="txt" href="#E6:118_4_1_2_1"><span class="lab"><font color="Green" title="E117">A46</font></span></a>, <a class="txt" href="#E27:118_4_1_2_1"><span class="lab"><font color="Green" title="E129">A58</font></span></a>, <a class="txt" href="#E33:118_4_1_2_1"><span class="lab"><font color="Green" title="E133">A70</font></span></a>, <a class="ref" href="group_2.html#T114" title="GROUP_2:th.114">GROUP_2:114</a></span>;<br/><a NAME="E36:118_4_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">a1</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="c26">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</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="E37:118_4_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">a1</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="c26">a29</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</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="E38:118_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A71</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">a1</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="c26">a29</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E25"><span class="lab"><font color="Green" title="E24">Lm10</font></span></a></span>;<br/><a NAME="E39:118_4_1_2_1"/><span class="lab"><font color="Green" title="E135">A72</font></span>: 
<font color="Maroon" title="c29">b1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">a1</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="txt" href="#E26:118_4_1_2_1"><span class="lab"><font color="Green" title="E128">A57</font></span></a>, <a class="ref" href="group_6.html#T32" title="GROUP_6:th.32">GROUP_6:32</a></span>;<br/><font color="Maroon" title="c27">b29</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:118_4_1_2_1"><span class="lab"><font color="Green" title="E124">A53</font></span></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">a29</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><a NAME="E41:118_4_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c29">b1</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="c27">b29</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:118_4_1_2_1"><span class="lab"><font color="Green" title="E135">A72</font></span></a>, <a class="txt" href="#E38:118_4_1_2_1"><span class="lab"><font color="Green" title="E134">A71</font></span></a>, <a class="ref" href="group_6.html#D6" title="GROUP_6:def.6">GROUP_6:def 6</a></span>;<br/><a NAME="E42:118_4_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c29">b1</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="c27">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</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="E43:118_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A73</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">b1</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="c27">b29</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">M9</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="E44:118_4_1_2_1"/>
<font color="Maroon" title="c22">B19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">b1</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:118_4_1_2_1"><span class="lab"><font color="Green" title="E127">A56</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E45:118_4_1_2_1"/>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:118_4_1_2_1"><span class="lab"><font color="Green" title="E132">A65</font></span></a>, <a class="txt" href="#E30:118_4_1_2_1"><span class="lab"><font color="Green" title="E131">A64</font></span></a>, <a class="txt" href="#E29:118_4_1_2_1"><span class="lab"><font color="Green" title="E130">A63</font></span></a>, <a class="txt" href="#E43:118_4_1_2_1"><span class="lab"><font color="Green" title="E136">A73</font></span></a>, <a class="txt" href="#E20:118_4_1_2_1"><span class="lab"><font color="Green" title="E125">A54</font></span></a>, <a class="ref" href="group_2.html#T114" title="GROUP_2:th.114">GROUP_2:114</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font> = 
 the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_2"><span class="lab"><font color="Green" title="E105">A34</font></span></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:118_4_1_2"><span class="lab"><font color="Green" title="E109">A38</font></span></a>, <a class="txt" href="#E13:118_4_1_2"><span class="lab"><font color="Green" title="E111">A40</font></span></a>, <a class="txt" href="#E15:118_4_1_2"><span class="lab"><font color="Green" title="E113">A42</font></span></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_4_1_2"><span class="lab"><font color="Green" title="E105">A34</font></span></a>, <a class="txt" href="#E4:118_4_1_2"><span class="lab"><font color="Green" title="E106">A35</font></span></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
<a NAME="E18:118"/>
 <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c8">MN99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118"><span class="lab"><font color="Green" title="E99">A4</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> =  <a href="group_6.html#K4" title="GROUP_6:func.4">CosOp</a> <font color="Maroon" title="c8">MN99</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">W1</font>, <font color="Maroon" title="c13">W2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c9">MN9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">N</font>  st <font color="Maroon" title="c12">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c13">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A2</font> holds <br/><font color="Maroon" title="c11">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c12">W1</font>,<font color="Maroon" title="c13">W2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Olive" title="b2">A2</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c14">W19</font> = <font color="Maroon" title="c12">W1</font>, <font color="Maroon" title="c15">W29</font> = <font color="Maroon" title="c13">W2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c8">MN99</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118"><span class="lab"><font color="Green" title="E99">A4</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c16">A1</font>, <font color="Maroon" title="c17">A2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">A1</font> &amp; <font color="Maroon" title="c13">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">A2</font> implies <font color="Maroon" title="c11">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c12">W1</font>,<font color="Maroon" title="c13">W2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font> )</span><br/><span class="kw">assume </span><a NAME="E2:118_5"/><span class="lab"><font color="Green" title="E105">A74</font></span>: 
<font color="Maroon" title="c12">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">A1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">A2</font> implies <font color="Maroon" title="c11">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c12">W1</font>,<font color="Maroon" title="c13">W2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c18">A19</font> = <font color="Maroon" title="c16">A1</font>, <font color="Maroon" title="c19">A29</font> = <font color="Maroon" title="c17">A2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">N9</font> ;<br/><span class="kw">assume </span><a NAME="E4:118_5"/><span class="lab"><font color="Green" title="E106">A75</font></span>: 
<font color="Maroon" title="c13">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c12">W1</font>,<font color="Maroon" title="c13">W2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font></span><br/><div><span class="lab"><font color="Green" title="E107">A76</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font> implies <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A19</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c19">A29</font> ) &amp; ( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A19</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c19">A29</font> implies <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A19</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c19">A29</font> implies <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:118_5_1_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A19</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c19">A29</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">g</font>, <font color="Maroon" title="c22">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E3:118_5_1_1"/><span class="lab"><font color="Green" title="E108">A77</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c22">h</font>
 <span class="kw">and </span><br/><a NAME="E4:118_5_1_1"/><span class="lab"><font color="Green" title="E109">A78</font></span>: 
( <font color="Maroon" title="c21">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A1</font> &amp; <font color="Maroon" title="c22">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A2</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">g9</font> = <font color="Maroon" title="c21">g</font>, <font color="Maroon" title="c24">h9</font> = <font color="Maroon" title="c22">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font> ;<br/><a NAME="E6:118_5_1_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c24">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_5_1_1"><span class="lab"><font color="Green" title="E108">A77</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:118_5_1_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A19</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c19">A29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_5_1_1"><span class="lab"><font color="Green" title="E109">A78</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:118_5_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A19</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c19">A29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">g9</font>, <font color="Maroon" title="c22">h9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">N9</font><span class="kw"> such that </span><br/><a NAME="E4:118_5_1"/><span class="lab"><font color="Green" title="E108">A79</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c22">h9</font>
 <span class="kw">and </span><br/><a NAME="E5:118_5_1"/><span class="lab"><font color="Green" title="E109">A80</font></span>: 
( <font color="Maroon" title="c21">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A19</font> &amp; <font color="Maroon" title="c22">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">A29</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">g</font> = <font color="Maroon" title="c21">g9</font>, <font color="Maroon" title="c24">h</font> = <font color="Maroon" title="c22">h9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><a NAME="E7:118_5_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c24">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_5_1"><span class="lab"><font color="Green" title="E108">A79</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:118_5_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:118_5_1"><span class="lab"><font color="Green" title="E109">A80</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><font color="Maroon" title="c11">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c12">W1</font>,<font color="Maroon" title="c13">W2</font>) = 
<font color="Maroon" title="c11">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c14">W19</font>,<font color="Maroon" title="c15">W29</font>)

<br/>.= 
<font color="Maroon" title="c18">A19</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c19">A29</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118_5"><span class="lab"><font color="Green" title="E105">A74</font></span></a>, <a class="txt" href="#E4:118_5"><span class="lab"><font color="Green" title="E106">A75</font></span></a>, <a class="ref" href="group_6.html#D3" title="GROUP_6:def.3">GROUP_6:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c16">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c17">A2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:118_5"><span class="lab"><font color="Green" title="E107">A76</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c10">K</font> = 
 <a href="group_6.html#K4" title="GROUP_6:func.4">CosOp</a> <font color="Maroon" title="c9">MN9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D3" title="GROUP_6:def.3">GROUP_6:def 3</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></span>
;<br/>
<span class="kw">then </span> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c5">M9</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:118"><span class="lab"><font color="Green" title="E100">A15</font></span></a>, <a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></span>
;<br/>
<a NAME="E23:118"/><span class="kw">then </span>
<font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:118"><span class="lab"><font color="Green" title="E103">A24</font></span></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:118"/>
<font color="Maroon" title="c4">N</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">MN</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:118"><span class="lab"><font color="Green" title="E101">A16</font></span></a>, <a class="txt" href="#E17:118"><span class="lab"><font color="Green" title="E104">A25</font></span></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a>, <a class="ref" href="group_9.html#D10" target="_self" title="GROUP_9:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
