<?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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span></span><br/>







<span class="kw">thus </span><a NAME="E1:143"/>
<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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:143_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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_2.html#K14" title="GROUP_2:func.14">*</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</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">g</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="E3:143_1"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E4:143_1"/><span class="lab"><font color="Green" title="E69">A2</font></span>: 
<font color="Maroon" title="c6">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_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_2.html#T126" target="_self" title="GROUP_2:th.126">Th126</a></span>;<br/>
<a NAME="E5:143_1"/>
<font color="Maroon" title="c6">g</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="#E4:143_1"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="group_2.html#T99" target="_self" title="GROUP_2:th.99">Th99</a></span>;<br/>
<a NAME="E6:143_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:143_1"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="group_2.html#T126" target="_self" title="GROUP_2:th.126">Th126</a></span>;<br/>
<a NAME="E7:143_1"/>
<font color="Maroon" title="c6">g</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="#E4:143_1"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="group_2.html#T99" target="_self" title="GROUP_2:th.99">Th99</a></span>;<br/>
<a NAME="E8:143_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:143_1"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="group_2.html#T126" target="_self" title="GROUP_2:th.126">Th126</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:143_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:143_1"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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">let </span><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:143"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E3:143"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">g</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="E5:143"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E6:143"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<font color="Maroon" title="c6">g</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_2.html#T126" target="_self" title="GROUP_2:th.126">Th126</a></span>;<br/>
<a NAME="E7:143"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H2</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:143"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">g1</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="E9:143"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E10:143"/><span class="lab"><font color="Green" title="E72">A8</font></span>: 
<font color="Maroon" title="c7">g1</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_2.html#T126" target="_self" title="GROUP_2:th.126">Th126</a></span>;<br/>
<a NAME="E11:143"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:143"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E9:143"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="group_1.html#T14" title="GROUP_1:th.14">GROUP_1:14</a></span>;<br/>
<a NAME="E12:143"/><span class="kw">then </span>
<font color="Maroon" title="c6">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_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="#E6:143"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E10:143"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="ref" href="group_2.html#T99" target="_self" title="GROUP_2:th.99">Th99</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:143"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:143"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="group_2.html#T126" target="_self" title="GROUP_2:th.126">Th126</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>
