<?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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Olive" title="b3">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Olive" title="b1">a</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Olive" title="b1">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</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"> for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Olive" title="b2">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">H1</font>, <font color="Maroon" title="c4">H2</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_2.html#D6" title="GROUP_2:def.6">GROUP_2:def 6</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="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> ) &amp; (  not <font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> or <font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font> ) )</span><br/>









<span class="kw">thus </span><a NAME="E1:73"/>
( <font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> )
  <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="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> or <font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:73_1"/>
<font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</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="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">h</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="E3:73_1"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c6">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></span>;<br/>
<a NAME="E5:73_1"/>
<font color="Maroon" title="c6">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="group_2.html#T99" title="GROUP_2:th.99">GROUP_2:99</a></span>;<br/>
<a NAME="E6:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></span>;<br/>
<a NAME="E7:73_1"/>
<font color="Maroon" title="c6">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="group_2.html#T99" title="GROUP_2:th.99">GROUP_2:99</a></span>;<br/>
<a NAME="E8:73_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:73_1"/>
<font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="group_2.html#T99" title="GROUP_2:th.99">GROUP_2:99</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">assume </span><a NAME="E2:73"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E3:73"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T99" title="GROUP_2:th.99">GROUP_2:99</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">b</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="E5:73"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E6:73"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></span>;<br/>
<a NAME="E7:73"/>
<font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="group_2.html#T99" title="GROUP_2:th.99">GROUP_2:99</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">c</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="E9:73"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E10:73"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></span>;<br/>
<a NAME="E11:73"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E9:73"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="group_3.html#T21" target="_self" title="GROUP_3:th.21">Th21</a></span>;<br/>
<a NAME="E12:73"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E10:73"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="group_2.html#T99" title="GROUP_2:th.99">GROUP_2:99</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:73"/>
<font color="Maroon" title="c5">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</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>
