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<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">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <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>;<br/><b>let </b><font color="Maroon">H</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">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>







<b>assume </b><a NAME="E1:8"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/>( <font color="Olive">e</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">X1</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ex <font color="Olive">Y1</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="mcart_1.html#K12">:]</a></span> )
 ;<br/>

<b>let </b><font color="Maroon">u</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="E2:8"/><i><font color="Green">E29</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"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E3:8"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p3">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p3">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:8"/><i><font color="Green">E32</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">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:8"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">Y1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E9:8"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</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>, </i>;<br/>
<a NAME="E10:8"/><i><font color="Green">E36</font></i>: 
<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="funct_3.html#T9">FUNCT_3:9</a></i>;<br/>
<a NAME="E11:8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</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>
 
<b>by </b><i/>;<br/>
<a NAME="E12:8"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>;<br/>
<a NAME="E13:8"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D1">FUNCT_3:def 1</a></i>;<br/>
<a NAME="E14:8"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E15:8"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E16:8"/><b>then </b><i><font color="Green">E37</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> &amp; <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="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E17:8"/>
 ex <font color="Olive">u1</font>, <font color="Olive">u2</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">u1</font>,<font color="Olive">u2</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E18:8"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T11">MCART_1:11</a></i>;<br/>
<a NAME="E19:8"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E20:8"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


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