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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[$1,$2 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,$2 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ];<br/>
<a NAME="E1:7_1"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  ex <font color="Olive" title="b2">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>consider </b><font color="Maroon" title="c2">ai</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon" title="c3">bi</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E2:7_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font>]
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<b>take </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
;<br/>

<b>thus </b><a NAME="E3:7_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>

<a NAME="E4:7_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ai</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">bi</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:7_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ]
 <b>by </b><i><a class="txt" href="jct_misc.html#E2:7_1_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green" title="E8">A4</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="jct_misc.html#E1:7_1"><i><font color="Green" title="E7">A2</font></i></a>);<br/></i>
<b>take </b>
 <a href="funct_2.html#K5" title="FUNCT_2:func.5">pr1</a> <font color="Maroon" title="c1">f</font>
;<br/>

<b>take </b>
 <a href="funct_2.html#K6" title="FUNCT_2:func.6">pr2</a> <font color="Maroon" title="c1">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E4:7_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr1</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K6" title="FUNCT_2:func.6">pr2</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D6" title="FUNCT_2:def.6">FUNCT_2:def 6</a>, <a class="ref" href="funct_2.html#D7" title="FUNCT_2:def.7">FUNCT_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:7_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr1</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K6" title="FUNCT_2:func.6">pr2</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font>]
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:7_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>


</div>
