<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">D</font> = <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">a<sub>2</sub></font>];<br/>
<a NAME="E1:6_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</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="E3:6_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>and </b><br/><a NAME="E4:6_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:6_1_1"/>
( ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">y1</font>] ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">y2</font>] ) )
 ;<br/>

<a NAME="E6:6_1_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:6_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>and </b><br/><a NAME="E4:6_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</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/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:6_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<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>assume </b><a NAME="E7:6_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E8:6_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E9:6_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:6_1"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E5:6_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E6:6_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/>

<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>assume </b><a NAME="E7:6_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 ;<br/>

<a NAME="E8:6_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:6_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>


</div>
