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

<b>let </b><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">Y1</font>, <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">Y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:118"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c1">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:118"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:118"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon" title="c2">Y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:118"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E7:118"/><i><font color="Green" title="E60">A5</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">Y1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <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" title="c3">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<a NAME="E8:118"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E1:118"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="zfmisc_1.html#E4:118"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T113" target="_self" title="ZFMISC_1:th.113">Th113</a></i>;<br/>
<a NAME="E9:118"/><b>then </b><i><font color="Green" title="E61">A6</font></i>: 
( <font color="Maroon" title="c3">X2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113" target="_self" title="ZFMISC_1:th.113">Th113</a></i>;<br/>
<a NAME="E10:118"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> iff <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:118_1"/>
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:118_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font>
 ;<br/>

<a NAME="E2:118_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E6:118"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="zfmisc_1.html#E7:118"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="zfmisc_1.html#E55"><i><font color="Green" title="E28">Lm18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:118_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E55"><i><font color="Green" title="E28">Lm18</font></i></a></i>;<br/>


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

<b>consider </b><font color="Maroon" title="c8">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:118_1"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E9:118"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:118_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font>
 ;<br/>

<a NAME="E5:118_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E3:118_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="zfmisc_1.html#E55"><i><font color="Green" title="E28">Lm18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:118_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E7:118"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="zfmisc_1.html#E55"><i><font color="Green" title="E28">Lm18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:118"/>
<font color="Maroon" title="c1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>

<a NAME="E12:118"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font> iff <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:118_2"/>
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:118_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font>
 ;<br/>

<a NAME="E2:118_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E3:118"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="zfmisc_1.html#E7:118"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="zfmisc_1.html#E55"><i><font color="Green" title="E28">Lm18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:118_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E55"><i><font color="Green" title="E28">Lm18</font></i></a></i>;<br/>


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

<b>consider </b><font color="Maroon" title="c8">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:118_2"/><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E9:118"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:118_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 ;<br/>

<a NAME="E5:118_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">z</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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E3:118_2"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="zfmisc_1.html#E55"><i><font color="Green" title="E28">Lm18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:118_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y1</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E7:118"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="zfmisc_1.html#E55"><i><font color="Green" title="E28">Lm18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:118"/>
<font color="Maroon" title="c2">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
