<?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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">N</font> being   <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> holds <br/> ( <font color="Olive" title="b1">B</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Olive" title="b2">N</font> is   <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="Olive" title="b1">B</font> &amp; <font color="Olive" title="b2">N</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Olive" title="b1">B</font> is   <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="Olive" title="b1">B</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N</font> being   <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> holds <br/> ( <font color="Maroon" title="c2">B</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Olive" title="b1">N</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b1">N</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c2">B</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</font> be   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">B</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c3">N</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c3">N</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c2">B</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:12"/>
<font color="Maroon" title="c2">B</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c3">N</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">N</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c2">B</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c2">B</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c3">N</font> as    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T88" onmouseover="rs('group_2/T88')" onmouseout="rh()">GROUP_2:88</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c2">B</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="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">b</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c5">b</font></span><br/><span class="kw">thus </span><a NAME="E1:12_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c5">b</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">A</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c5">b</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font><span class="kw"> such that </span><br/><a NAME="E3:12_1_1_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">and </span><br/><a NAME="E4:12_1_1_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T103" onmouseover="rs('group_2/T103')" onmouseout="rh()">GROUP_2:103</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">a9</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c9">b9</font> = <font color="Maroon" title="c5">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T42" onmouseover="rs('group_2/T42')" onmouseout="rh()">GROUP_2:42</a></span>;<br/>
<a NAME="E6:12_1_1_1"/>
( <font color="Maroon" title="c7">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E4:12_1_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="group_2.html#T43" onmouseover="rs('group_2/T43')" onmouseout="rh()">GROUP_2:43</a>, <a class="ref" href="group_2.html#T82" onmouseover="rs('group_2/T82')" onmouseout="rh()">GROUP_2:82</a></span>;<br/>
<a NAME="E7:12_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T103" onmouseover="rs('group_2/T103')" onmouseout="rh()">GROUP_2:103</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x9</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a></span>;<br/>
<a NAME="E9:12_1_1_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T48" onmouseover="rs('group_2/T48')" onmouseout="rh()">GROUP_2:48</a></span>;<br/>
<a NAME="E10:12_1_1_1"/>
<font color="Maroon" title="c9">b9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T118" onmouseover="rs('group_3/T118')" onmouseout="rh()">GROUP_3:118</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b1</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="E12:12_1_1_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">b9</font>
 <span class="kw">and </span><br/><a NAME="E13:12_1_1_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c11">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_1_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="group_2.html#T104" onmouseover="rs('group_2/T104')" onmouseout="rh()">GROUP_2:104</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x99</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12_1_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a></span>;<br/>
<a NAME="E15:12_1_1_1"/>
<font color="Maroon" title="c11">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b9</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="#E12:12_1_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="group_1.html#T14" onmouseover="rs('group_1/T14')" onmouseout="rh()">GROUP_1:14</a></span>;<br/>
<a NAME="E16:12_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c11">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="txt" href="#E9:12_1_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="group_2.html#T43" onmouseover="rs('group_2/T43')" onmouseout="rh()">GROUP_2:43</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">b19</font> = <font color="Maroon" title="c11">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> ;<br/>
<a NAME="E18:12_1_1_1"/>
<font color="Maroon" title="c11">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:12_1_1_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a></span>;<br/>
<a NAME="E19:12_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c13">b19</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:12_1_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="group_2.html#T82" onmouseover="rs('group_2/T82')" onmouseout="rh()">GROUP_2:82</a></span>;<br/>
<a NAME="E20:12_1_1_1"/>
<font color="Maroon" title="c13">b19</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12_1_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="group_2.html#T43" onmouseover="rs('group_2/T43')" onmouseout="rh()">GROUP_2:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:12_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:12_1_1_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="group_2.html#T104" onmouseover="rs('group_2/T104')" onmouseout="rh()">GROUP_2:104</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E3:12_1"/>
<font color="Maroon" title="c2">B</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c3">N</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T118" onmouseover="rs('group_3/T118')" onmouseout="rh()">GROUP_3:118</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:12"/>
<font color="Maroon" title="c3">N</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c2">B</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
