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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Y1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>







<b>assume </b><a NAME="E1:6"/>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>
<a NAME="E3:6"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_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="E1:6_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<a NAME="E4:6_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E5:6_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E6:6_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:6_1_1"/>
( <font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>hence </b><a NAME="E8:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i>, <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a>, </i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y1</font>;<br/><b>assume </b><a NAME="E3:6_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><a NAME="E4:6_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:6_1"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:6_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/><b>hence </b><a NAME="E7:6_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:6"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:6_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<a NAME="E4:6_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E5:6_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E6:6_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:6_2_1"/>
( <font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>hence </b><a NAME="E8:6_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>by </b><i>, <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a>, </i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font>;<br/><b>assume </b><a NAME="E3:6_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 ;<br/><a NAME="E4:6_2"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:6_2"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:6_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>;<br/><b>hence </b><a NAME="E7:6_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:6"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
