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<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>, <font color="Olive" title="b2">H</font> being   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a>  holds <br/><font color="Olive" title="b2">H</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <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"> ( <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a>  implies <font color="Maroon" title="c3">H</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:141"/><span class="lab"><font color="Green" title="E118">A1</font></span>: 
<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 ; <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="c2">G</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a>  or <font color="Maroon" title="c3">H</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:141"/><span class="lab"><font color="Green" title="E119">A2</font></span>: 
<font color="Maroon" title="c2">G</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</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="c3">H</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:141"/><span class="lab"><font color="Green" title="E120">A3</font></span>: 
 not <font color="Maroon" title="c3">H</font> is  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_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="E1:141_1"/>
( <font color="Maroon" title="c3">H</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  or  ex <font color="Olive" title="b1">H9</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="c3">H</font> st <br/>( <font color="Olive" title="b1">H9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Olive" title="b1">H9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c3">H</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:141"><span class="lab"><font color="Green" title="E120">A3</font></span></a>, <a class="ref" href="group_9.html#D13" target="_self" title="GROUP_9:def.13">Def13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1"><span class="kw">suppose </span></a><a NAME="E1:141_1_1"/>
<font color="Maroon" title="c3">H</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:141_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">G</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141"><span class="lab"><font color="Green" title="E118">A1</font></span></a>, <a class="ref" href="group_9.html#T58" target="_self" title="GROUP_9:th.58">Th58</a></span>;<br/><span class="kw">hence </span><a NAME="E3:141_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:141"><span class="lab"><font color="Green" title="E119">A2</font></span></a>, <a class="ref" href="group_9.html#D13" target="_self" title="GROUP_9:def.13">Def13</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="141_1_2"><span class="kw">suppose </span></a><a NAME="E1:141_1_2"/>
 ex <font color="Olive" title="b1">H9</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="c3">H</font> st <br/>( <font color="Olive" title="b1">H9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Olive" title="b1">H9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c3">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">H9</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="c3">H</font><span class="kw"> such that </span><br/><a NAME="E3:141_1_2"/><span class="lab"><font color="Green" title="E121">A4</font></span>: 
<font color="Maroon" title="c4">H9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">and </span><br/><a NAME="E4:141_1_2"/><span class="lab"><font color="Green" title="E122">A5</font></span>: 
<font color="Maroon" title="c4">H9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c3">H</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font><span class="kw"> such that </span><br/><a NAME="E6:141_1_2"/><span class="lab"><font color="Green" title="E123">A6</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141"><span class="lab"><font color="Green" title="E118">A1</font></span></a>, <a class="ref" href="group_9.html#D19" target="_self" title="GROUP_9:def.19">Def22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">H99</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">H9</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">H9</font> #) as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H</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/><a NAME="E8:141_1_2"/>
 <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">H9</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">H9</font> #) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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 <font color="Maroon" title="c3">H</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">H</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:141_1_2"><span class="lab"><font color="Green" title="E121">A4</font></span></a>, <a class="txt" href="group_9.html#E12"><span class="lab"><font color="Green" title="E12">Lm5</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font><span class="kw"> such that </span><br/><a NAME="E10:141_1_2"/><span class="lab"><font color="Green" title="E124">A7</font></span>: 
 not <font color="Maroon" title="c7">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T62" title="GROUP_2:th.62">GROUP_2:62</a></span>;<br/><a NAME="E11:141_1_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:141_1_2"><span class="lab"><font color="Green" title="E122">A5</font></span></a>, <a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:141_1_2"/><span class="lab"><font color="Green" title="E125">A8</font></span>: 
<font color="Maroon" title="c8">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 <font color="Maroon" title="c4">H9</font>
 <span class="kw">and </span><br/><a NAME="E14:141_1_2"/><span class="lab"><font color="Green" title="E126">A9</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T35" title="ZFMISC_1:th.35">ZFMISC_1:35</a></span>;<br/><a NAME="E15:141_1_2"/><span class="lab"><font color="Green" title="E127">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:141_1_2"><span class="lab"><font color="Green" title="E125">A8</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E16:141_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T40" title="GROUP_2:th.40">GROUP_2:40</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">y</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="E19:141_1_2"/><span class="lab"><font color="Green" title="E128">A11</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:141_1_2"><span class="lab"><font color="Green" title="E123">A6</font></span></a>, <a class="ref" href="group_9.html#T52" target="_self" title="GROUP_9:th.52">Th52</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">A</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> : <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font> </span> } </span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">g</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="E21:141_1_2"/><span class="lab"><font color="Green" title="E129">A12</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:141_1_2"><span class="lab"><font color="Green" title="E123">A6</font></span></a>, <a class="ref" href="group_9.html#T52" target="_self" title="GROUP_9:th.52">Th52</a></span>;<br/><a NAME="E22:141_1_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" title="GROUP_2:th.46">GROUP_2:46</a></span>;<br/><a NAME="E23:141_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E33"><span class="lab"><font color="Green" title="E31">Lm13</font></span></a></span>;<br/><a NAME="E24:141_1_2"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> : <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font> </span> } </span> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">A</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> : <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font> </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_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> <font color="Maroon" title="c13">A</font> implies <font color="Maroon" title="c14">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 <font color="Maroon" title="c2">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:141_1_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</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="c14">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 <font color="Maroon" title="c2">G</font></span><br/><a NAME="E2:141_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g</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="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:141_1_2_1"/>
<font color="Maroon" title="c14">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 <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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">A</font> = <font color="Maroon" title="c13">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</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>;<br/><div><span class="lab"><font color="Green" title="E130">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">g1</font>, <font color="Maroon" title="c16">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( <font color="Maroon" title="c15">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c16">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> implies <font color="Maroon" title="c15">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:141_1_2_2"/><span class="lab"><font color="Green" title="E131">A14</font></span>: 
<font color="Maroon" title="c15">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <span class="kw">and </span><br/><a NAME="E2:141_1_2_2"/><span class="lab"><font color="Green" title="E132">A15</font></span>: 
<font color="Maroon" title="c16">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</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="c15">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c17">b</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="E4:141_1_2_2"/><span class="lab"><font color="Green" title="E133">A16</font></span>: 
<font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g2</font>
 <span class="kw">and </span><br/><a NAME="E5:141_1_2_2"/><span class="lab"><font color="Green" title="E134">A17</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:141_1_2_2"><span class="lab"><font color="Green" title="E132">A15</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">a</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="E7:141_1_2_2"/><span class="lab"><font color="Green" title="E135">A18</font></span>: 
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g1</font>
 <span class="kw">and </span><br/><a NAME="E8:141_1_2_2"/><span class="lab"><font color="Green" title="E136">A19</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141_1_2_2"><span class="lab"><font color="Green" title="E131">A14</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">fb</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c20">fa</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font>;<br/><a NAME="E9:141_1_2_2"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:141_1_2_2"><span class="lab"><font color="Green" title="E136">A19</font></span></a>, <a class="txt" href="#E5:141_1_2_2"><span class="lab"><font color="Green" title="E134">A17</font></span></a>, <a class="ref" href="group_2.html#T50" title="GROUP_2:th.50">GROUP_2:50</a>, <a class="ref" href="group_6.html#D6" title="GROUP_6:def.6">GROUP_6:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E10:141_1_2_2"/>
<font color="Maroon" title="c15">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:141_1_2_2"><span class="lab"><font color="Green" title="E135">A18</font></span></a>, <a class="txt" href="#E4:141_1_2_2"><span class="lab"><font color="Green" title="E133">A16</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><div><span class="lab"><font color="Green" title="E131">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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">  for <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>  st <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> holds <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="c15">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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></span><br/><span class="kw">let </span><font color="Maroon" title="c16">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( <font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> implies <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="c15">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:141_1_2_3"/>
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</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="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c15">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">a</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="E3:141_1_2_3"/><span class="lab"><font color="Green" title="E132">A21</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font>
 <span class="kw">and </span><br/><a NAME="E4:141_1_2_3"/><span class="lab"><font color="Green" title="E133">A22</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 ;<br/><a NAME="E5:141_1_2_3"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">a</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="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:141_1_2_3"><span class="lab"><font color="Green" title="E133">A22</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E6:141_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H9</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="E7:141_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c15">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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:141_1_2_3"><span class="lab"><font color="Green" title="E132">A21</font></span></a>, <a class="txt" href="group_9.html#E25"><span class="lab"><font color="Green" title="E24">Lm10</font></span></a></span>;<br/><a NAME="E8:141_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <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="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c15">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D18" target="_self" title="GROUP_9:def.18">Def18</a></span>;<br/><a NAME="E9:141_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <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="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c15">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <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">H9</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="E10:141_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <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="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c15">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E11:141_1_2_3"/>
<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="c15">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 ; <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="hs2(this)" href="javascript:()" title="141_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( <font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> implies <font color="Maroon" title="c15">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:141_1_2_4"/>
<font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</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="c15">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">a</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="E3:141_1_2_4"/><span class="lab"><font color="Green" title="E132">A23</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">and </span><br/><a NAME="E4:141_1_2_4"/><span class="lab"><font color="Green" title="E133">A24</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 ;<br/><a NAME="E5:141_1_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:141_1_2_4"><span class="lab"><font color="Green" title="E133">A24</font></span></a>, <a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></span>;<br/><a NAME="E6:141_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E34"><span class="lab"><font color="Green" title="E32">Lm14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:141_1_2_4"/>
<font color="Maroon" title="c15">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:141_1_2_4"><span class="lab"><font color="Green" title="E132">A23</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">then </span><span class="kw">consider </span><font color="Maroon" title="c15">G99</font> being   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E32:141_1_2"/><span class="lab"><font color="Green" title="E132">A25</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">G99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:141_1_2"><span class="lab"><font color="Green" title="E130">A13</font></span></a>, <a class="txt" href="#E29:141_1_2"><span class="lab"><font color="Green" title="E131">A20</font></span></a>, <a class="txt" href="group_9.html#E35"><span class="lab"><font color="Green" title="E33">Lm15</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">G9</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="c15">G99</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c15">G99</font> #) as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E37"><span class="lab"><font color="Green" title="E35">Lm16</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> <font color="Maroon" title="c17">g</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c16">G9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">G9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c17">g</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">g</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c16">G9</font> implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">G9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c17">g</font> )</span><br/><a NAME="E1:141_1_2_5_1"/><span class="lab"><font color="Green" title="E133">A26</font></span>: 
<font color="Maroon" title="c6">H99</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D13" title="GROUP_3:def.13">GROUP_3:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:141_1_2_5_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">g</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c16">G9</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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">G9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c17">g</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">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="E4:141_1_2_5_1"/><span class="lab"><font color="Green" title="E134">A27</font></span>: 
<font color="Maroon" title="c18">x</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="c19">h</font>
 <span class="kw">and </span><br/><a NAME="E5:141_1_2_5_1"/><span class="lab"><font color="Green" title="E135">A28</font></span>: 
<font color="Maroon" title="c19">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:141_1_2"><span class="lab"><font color="Green" title="E132">A25</font></span></a>, <a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c20">h9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>;<br/><span class="lab"><font color="Green" title="E136">A29</font></span>: <font color="Maroon" title="c5">f</font> <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="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" title="GROUP_6:def.6">GROUP_6:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" title="GROUP_6:def.6">GROUP_6:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font></span>)</span> <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="group_9.html#E34"><span class="lab"><font color="Green" title="E32">Lm14</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D2" title="GROUP_3:def.2">GROUP_3:def 2</a></span>
;<br/><a NAME="E7:141_1_2_5_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:141_1_2_5_1"><span class="lab"><font color="Green" title="E135">A28</font></span></a></span>;<br/><a NAME="E8:141_1_2_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <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="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141_1_2_5_1"><span class="lab"><font color="Green" title="E133">A26</font></span></a>, <a class="txt" href="#E6:141_1_2_5_1"><span class="lab"><font color="Green" title="E136">A29</font></span></a>, <a class="ref" href="group_3.html#T58" title="GROUP_3:th.58">GROUP_3:58</a></span>;<br/><a NAME="E9:141_1_2_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">g</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="c17">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:141_1_2_5_1"><span class="lab"><font color="Green" title="E134">A27</font></span></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:141_1_2_5_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">G9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:141_1_2"><span class="lab"><font color="Green" title="E132">A25</font></span></a>, <a class="txt" href="#E9:141_1_2_5_1"><span class="lab"><font color="Green" title="E137">A30</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="E2:141_1_2_5"/>
<font color="Maroon" title="c17">g</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c16">G9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">G9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c17">g</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><a NAME="E35:141_1_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">H</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">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 <font color="Maroon" title="c15">G99</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c15">G99</font> #) holds <br/><font color="Olive" title="b1">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>;<br/><a NAME="E36:141_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A31</font></span>: 
<font color="Maroon" title="c15">G99</font> is  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D10" target="_self" title="GROUP_9:def.10">Def10</a></span>;<br/><a NAME="E37:141_1_2"/><span class="lab"><font color="Green" title="E134">A32</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:141_1_2"><span class="lab"><font color="Green" title="E126">A9</font></span></a>, <a class="txt" href="#E19:141_1_2"><span class="lab"><font color="Green" title="E128">A11</font></span></a>, <a class="txt" href="group_9.html#E33"><span class="lab"><font color="Green" title="E31">Lm13</font></span></a></span>;<br/><a NAME="E38:141_1_2"/>
<font color="Maroon" title="c10">y</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="c15">G99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:141_1_2"><span class="lab"><font color="Green" title="E132">A25</font></span></a>, <a class="txt" href="#E15:141_1_2"><span class="lab"><font color="Green" title="E127">A10</font></span></a>, <a class="txt" href="#E19:141_1_2"><span class="lab"><font color="Green" title="E128">A11</font></span></a></span>;<br/><a NAME="E39:141_1_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">G99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:141_1_2"><span class="lab"><font color="Green" title="E134">A32</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E40:141_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A33</font></span>: 
<font color="Maroon" title="c15">G99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:141_1_2_6"/>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:141_1_2_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g9</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">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g9</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H99</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:141_1_2_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:141_1_2"><span class="lab"><font color="Green" title="E124">A7</font></span></a>, <a class="txt" href="#E21:141_1_2"><span class="lab"><font color="Green" title="E129">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><a NAME="E42:141_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">G99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:141_1_2"><span class="lab"><font color="Green" title="E132">A25</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E43:141_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:141"><span class="lab"><font color="Green" title="E119">A2</font></span></a>, <a class="txt" href="#E40:141_1_2"><span class="lab"><font color="Green" title="E135">A33</font></span></a>, <a class="txt" href="#E36:141_1_2"><span class="lab"><font color="Green" title="E133">A31</font></span></a>, <a class="ref" href="group_9.html#D13" target="_self" title="GROUP_9:def.13">Def13</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>
