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

<b>set </b><font color="Maroon" title="c1">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u'</font>,<font color="Olive" title="b2">v'</font>) where <font color="Olive" title="b1">u'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span> ;<br/>
<a NAME="E1:36_1"/><i><font color="Green" title="E12">A3</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E1:36"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fraenkel.html#E2:36"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="finset_1.html#T19" title="FINSET_1:th.19">FINSET_1:19</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>);<br/>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green" title="E13">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E4:36_1"/><i><font color="Green" title="E14">A5</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E5:36_1"/><i><font color="Green" title="E15">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E3:36_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E6:36_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u'</font>,<font color="Olive" title="b2">v'</font>) where <font color="Olive" title="b1">u'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u'</font>,<font color="Olive" title="b2">v'</font>) where <font color="Olive" title="b1">u'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u'</font>,<font color="Olive" title="b2">v'</font>) where <font color="Olive" title="b1">u'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c5">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:36_1_1_1"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>)
 <b>and </b><br/><a NAME="E4:36_1_1_1"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() )
 ;<br/>
<a NAME="E5:36_1_1_1"/><i><font color="Green" title="E18">A9</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E4:36_1_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:36_1_1_1"/><b>then </b><i><font color="Green" title="E19">A10</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E5:36_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:36_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E8:36_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>)
 
<b>by </b><i><a class="txt" href="fraenkel.html#E3:36_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="fraenkel.html#E4:36_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="fraenkel.html#E6:36_1_1_1"><i><font color="Green" title="E19">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:36_1_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="fraenkel.html#E5:36_1_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="fraenkel.html#E6:36_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:36_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:36_1_1"/><i><font color="Green" title="E16">A11</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E5:36_1_1"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E6:36_1_1"/><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="txt" href="fraenkel.html#E5:36_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="fraenkel.html#E4:36_1_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:36_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E9:36_1_1"/><b>then </b><i><font color="Green" title="E19">A14</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon" title="c5">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() )
 
<b>by </b><i><a class="txt" href="fraenkel.html#E5:36_1_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:36_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>)
 
<b>by </b><i><a class="txt" href="fraenkel.html#E3:36_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="fraenkel.html#E4:36_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="fraenkel.html#E4:36_1_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="fraenkel.html#E6:36_1_1"><i><font color="Green" title="E18">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:36_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u'</font>,<font color="Olive" title="b2">v'</font>) where <font color="Olive" title="b1">u'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E9:36_1_1"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u'</font>,<font color="Olive" title="b2">v'</font>) where <font color="Olive" title="b1">u'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:36_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">FINSET_1:17</a></i>;<br/>


</div>
