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<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>let </b><font color="Maroon">A1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X1</font>;<br/><b>let </b><font color="Maroon">A2</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X2</font>;<br/>







<b>assume </b><a NAME="E1:60"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="mcart_1.html#K12">:]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X1</font>, <font color="Olive">b</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X2</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> ) </span>}</span> 
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A1</font>, <font color="Maroon">a2</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A2</font>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X1</font> ;<br/><b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a2</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X2</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><a NAME="E2:60_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E3:60_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:60"><i><font color="Green">E32</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>hence </b><a NAME="E4:60_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:60_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><a NAME="E8:60_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:60_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:60_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E3:60_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:60"><i><font color="Green">E32</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>hence </b><a NAME="E4:60_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:60_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><a NAME="E10:60_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:60_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>hence </b><a NAME="E11:60_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A1</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A2</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:60_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div>
<b>end;</b></div>

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

<b>assume </b><a NAME="E3:60"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A1</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A2</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:60"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X1</font>, <font color="Maroon">b</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X2</font><b> such that </b><br/><a NAME="E3:60_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="mcart_1.html#K12">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i/>;<br/><a NAME="E4:60_2"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>hence </b><a NAME="E5:60_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:60"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:60"><i><font color="Green">E32</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>


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