<?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>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span><br/>





<span class="kw">thus </span><a NAME="E1:43"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
  <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="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <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="E3:43_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E4:43_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 <span class="kw">and </span><br/><a NAME="E5:43_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/>
<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="E7:43_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<a NAME="E8:43_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  &amp; <font color="Maroon" title="c7">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="group_3.html#T32" target="_self" title="GROUP_3:th.32">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:43_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 ; <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="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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:43"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 ; <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</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="E4:43"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">and </span><br/><a NAME="E5:43"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">b</font>, <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="E7:43"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c5">a</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="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E8:43"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E9:43"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/>
<a NAME="E10:43"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c6">b</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="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E31">A8</font></span></a></span>;<br/>
<a NAME="E11:43"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="group_3.html#T32" target="_self" title="GROUP_3:th.32">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:43"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_3.html#K3" title="GROUP_3:func.3">|^</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E33">A10</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"> verum</span><br/>


</div>
