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

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v</font>] ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v</font>] ) </span>}</span>  implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v</font>] ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E4:1_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">v</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">v</font>] )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">v</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">v</font>] )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">v</font>]
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">v</font>]
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>assume </b><a NAME="E2:1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:1_1_1_2_1"/><b>then </b>
 ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v</font>] )
 ;<br/><b>hence </b><a NAME="E3:1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v</font>] ) </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:1_1_1_2_2"/><b>then </b>
 ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v</font>] )
 ;<br/><b>hence </b><a NAME="E3:1_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v</font>] ) </span>}</span> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v</font>] ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">v1</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">v2</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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