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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ];<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>);<br/>
<a NAME="E1:1_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)</span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>)</span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ] </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</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="E1:1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:1_1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>)</span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:1_1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="functor0.html#E1:1"><i><font color="Green">E1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>)</span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ] </span>}</span> 
 <b>by </b><i><a class="txt" href="functor0.html#E3:1_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="functor0.html#E4:1_1_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>1</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="E2:1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>)</span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E4:1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</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>2</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>)</span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> ]
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E7:1_1_1"/>
<font color="Maroon">c<sub>2</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></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/>
<b>hence </b><a NAME="E8:1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="functor0.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="functor0.html#E4:1_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="functor0.html#E5:1_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:1_1"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E4:1_1"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="functor0.html#E2:1"><i><font color="Green">E2</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() = 
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>

<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
<b>from </b><i><a class="ref" href="altcat_2.html#S3">ALTCAT_2:sch 3</a>(<a class="txt" href="functor0.html#E1:1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="functor0.html#E4:1_1"><i><font color="Green">E5</font></i></a>)</i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>())
<b>by </b><i><a class="txt" href="functor0.html#E3:1_1"><i><font color="Green">E4</font></i></a></i>
;<br/>


</div>
