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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span>;<br/>
<a NAME="E1:8"/><i><font color="Green">E5</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow12.html#T4" target="_self">Th4</a></i>;<br/>
<b>thus </b><a NAME="E2:8"/>
<span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow12.html#T5" target="_self">Th5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></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="E3:8"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:8"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:8"/><i><font color="Green">E7</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="yellow12.html#E5:8"><i><font color="Green">E6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:8"/>
<span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow12.html#E5:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="yellow12.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:8"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow12.html#E1:8"><i><font color="Green">E5</font></i></a>, <a class="txt" href="yellow12.html#E5:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="yellow12.html#E6:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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