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

<b>let </b><font color="Maroon" title="c9">f</font> be    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">R1</font>,<font color="Maroon" title="c5">S1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c6">S2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>thus </b><a NAME="E1:11_1_1"/>
( <font color="Maroon" title="c9">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c8">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span> implies for <font color="Olive" title="b1">r1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">R1</font><br/> for <font color="Olive" title="b2">r2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">R2</font><br/> for <font color="Olive" title="b3">s1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">S1</font><br/> for <font color="Olive" title="b4">s2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S2</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b3">s1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">s2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></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> <font color="Maroon" title="c9">f</font> iff ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</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> <font color="Maroon" title="c7">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">s1</font>,<font color="Olive" title="b4">s2</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> <font color="Maroon" title="c8">S</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c8">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c10">r1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">R1</font>;<br/><b>let </b><font color="Maroon" title="c11">r2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">R2</font>;<br/><b>let </b><font color="Maroon" title="c12">s1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">S1</font>;<br/><b>let </b><font color="Maroon" title="c13">s2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S2</font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c12">s1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">r2</font>,<font color="Maroon" title="c13">s2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></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> <font color="Maroon" title="c9">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">r1'</font>, <font color="Maroon" title="c15">s1'</font>, <font color="Maroon" title="c16">r2'</font>, <font color="Maroon" title="c17">s2'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:11_1_1_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c12">s1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">r1'</font>,<font color="Maroon" title="c15">s1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">r2</font>,<font color="Maroon" title="c13">s2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">r2'</font>,<font color="Maroon" title="c17">s2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E4:11_1_1_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c14">r1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp; <font color="Maroon" title="c15">s1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> &amp; <font color="Maroon" title="c16">r2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Maroon" title="c17">s2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font> &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">r1'</font>,<font color="Maroon" title="c16">r2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">s1'</font>,<font color="Maroon" title="c17">s2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> ) )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:11_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="pcs_0.html#D2" target="_self" title="PCS_0:def.2">Def2</a></i>;<br/><a NAME="E5:11_1_1_1_1"/>
( <font color="Maroon" title="c10">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r1'</font> &amp; <font color="Maroon" title="c11">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r2'</font> &amp; <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1'</font> &amp; <font color="Maroon" title="c13">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s2'</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:11_1_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:11_1_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="c10">r1</font>,<font color="Maroon" title="c11">r2</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> <font color="Maroon" title="c7">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">s1</font>,<font color="Maroon" title="c13">s2</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> <font color="Maroon" title="c8">S</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:11_1_1_1_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:11_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="c10">r1</font>,<font color="Maroon" title="c11">r2</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> <font color="Maroon" title="c7">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">s1</font>,<font color="Maroon" title="c13">s2</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> <font color="Maroon" title="c8">S</font> ) implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c12">s1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">r2</font>,<font color="Maroon" title="c13">s2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></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> <font color="Maroon" title="c9">f</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:11_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="pcs_0.html#D2" target="_self" title="PCS_0:def.2">Def2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:11_1_1"/><i><font color="Green" title="E5">A4</font></i>: 
for <font color="Olive" title="b1">r1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">R1</font><br/> for <font color="Olive" title="b2">r2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">R2</font><br/> for <font color="Olive" title="b3">s1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">S1</font><br/> for <font color="Olive" title="b4">s2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S2</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b3">s1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">s2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></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> <font color="Maroon" title="c9">f</font> iff ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</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> <font color="Maroon" title="c7">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">s1</font>,<font color="Olive" title="b4">s2</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> <font color="Maroon" title="c8">S</font> ) )
 ;<br/>

<a NAME="E3:11_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c8">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:11_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c8">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1_2_1"/><i><font color="Green" title="E6">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<a NAME="E2:11_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T20" title="RELAT_1:th.20">RELAT_1:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:11_1_1_2_1"/><i><font color="Green" title="E7">A6</font></i>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E5:11_1_1_2_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:11_1_1_2_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="relat_1.html#T20" title="RELAT_1:th.20">RELAT_1:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">y1</font>, <font color="Maroon" title="c15">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:11_1_1_2_1"/><i><font color="Green" title="E8">A7</font></i>: 
( <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Maroon" title="c15">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y1</font>,<font color="Maroon" title="c15">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E8:11_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c14">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x2</font>,<font color="Maroon" title="c15">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E2:11_1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="pcs_0.html#E1:11_1_1_2_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="pcs_0.html#E4:11_1_1_2_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="pcs_0.html#E7:11_1_1_2_1"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:11_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c8">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E4:11_1_1_2_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="pcs_0.html#E7:11_1_1_2_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="pcs_0.html#D2" target="_self" title="PCS_0:def.2">Def2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:11_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c8">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span>
 ;<br/>

<a NAME="E3:11_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">r2</font>, <font color="Olive" title="b4">s2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">r2</font>,<font color="Olive" title="b4">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp; <font color="Olive" title="b2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> &amp; <font color="Olive" title="b3">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Olive" title="b4">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font> &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b3">r2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">s1</font>,<font color="Olive" title="b4">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> ) )
 
<b>by </b><i><a class="ref" href="pcs_0.html#D2" target="_self" title="PCS_0:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:11_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:11_1_1"><i><font color="Green" title="E5">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:11_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c8">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a></i>;<br/>


</div>
