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

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">u1</font>,<font color="Olive">v1</font>) where <font color="Olive">u1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u1</font>,<font color="Olive">v1</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">u3</font>,<font color="Olive">v3</font>) where <font color="Olive">u3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u3</font>,<font color="Olive">v3</font>] </span>}</span> ;<br/>
<a NAME="E1:56_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">u1</font>,<font color="Olive">v1</font>) where <font color="Olive">u1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u1</font>,<font color="Olive">v1</font>] </span>}</span>  iff <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">u3</font>,<font color="Olive">v3</font>) where <font color="Olive">u3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u3</font>,<font color="Olive">v3</font>] </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:56_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">u1</font>,<font color="Olive">v1</font>) where <font color="Olive">u1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u1</font>,<font color="Olive">v1</font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:56_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">u</font>,<font color="Maroon">v</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">u</font>,<font color="Maroon">v</font>] )
 ;<br/><a NAME="E4:56_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">u</font>,<font color="Maroon">v</font>)
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:56_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">u3</font>,<font color="Olive">v3</font>) where <font color="Olive">u3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u3</font>,<font color="Olive">v3</font>] </span>}</span> 
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:56_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">u3</font>,<font color="Olive">v3</font>) where <font color="Olive">u3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u3</font>,<font color="Olive">v3</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E4:56_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">u</font>,<font color="Maroon">v</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">u</font>,<font color="Maroon">v</font>] )
 ;<br/>
<a NAME="E5:56_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">u</font>,<font color="Maroon">v</font>)
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:56_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">u1</font>,<font color="Olive">v1</font>) where <font color="Olive">u1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u1</font>,<font color="Olive">v1</font>] </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:56_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">u1</font>,<font color="Olive">v1</font>) where <font color="Olive">u1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u1</font>,<font color="Olive">v1</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">u2</font>,<font color="Olive">v2</font>) where <font color="Olive">u2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">v2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">u2</font>,<font color="Olive">v2</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
