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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">A</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> iff <span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">A</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> iff <span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">A</font> )</span><br/>



<span class="kw">thus </span><a NAME="E1:19"/>
( <font color="Maroon" title="c2">A</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> implies <span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="grnilp_1.html#E18"><span class="lab"><font color="Green" title="E11">Lm3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c2">A</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:19"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font></span><br/>

<a NAME="E3:19"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">b</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">Z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:19_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c6">Z</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></span>;<br/>
<a NAME="E5:19_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c6">Z</font> <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="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></span>;<br/>
<a NAME="E6:19_1"/>
<font color="Maroon" title="c3">b</font> <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>  <a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</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="E7:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Z</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">A</font></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="group_5.html#T74" title="GROUP_5:th.74">GROUP_5:74</a></span>;<br/>
<a NAME="E8:19_1"/>
<span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Z</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="group_2.html#T49" title="GROUP_2:th.49">GROUP_2:49</a></span>;<br/>
<a NAME="E9:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Z</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="c6">Z</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A7</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Z</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="c6">Z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</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="c3">b</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Z</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="c6">Z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</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="c3">b</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> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</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="c3">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</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="c3">b</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="c3">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</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="c3">b</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="c3">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</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="c1">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">Z</font>
<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/>
<span class="kw">hence </span><a NAME="E12:19_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E9:19_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E10:19_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:19"/>
<font color="Maroon" title="c2">A</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T141" title="GROUP_3:th.141">GROUP_3:141</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
