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<span class="kw">defpred </span><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> ] <span class="kw">means </span>for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,$2];<br/>
<a NAME="E1:52_1"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>] holds <br/><font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z</font>] implies <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() )</span><br/>



<span class="kw">assume </span><a NAME="E1:52_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z</font>] or <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() )</span><br/>

<a NAME="E2:52_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:52_1_1"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,<font color="Maroon" title="c2">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()</span><br/>

<span class="kw">hence </span><a NAME="E4:52_1_1"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:52_1_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:52_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">z1</font>, <font color="Olive" title="b3">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z2</font>] holds <br/><font color="Olive" title="b2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z2</font>] implies <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52_1_2"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E2:52_1_2"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
( ( for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,<font color="Maroon" title="c2">z1</font>] ) &amp; ( for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,<font color="Maroon" title="c3">z2</font>] ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:52_1_2"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 <span class="kw">and </span><br/><a NAME="E5:52_1_2"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E6:52_1_2"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c2">z1</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c3">z2</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E5:52_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:52_1_2"/>
<font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:52_1_2"><span class="lab"><font color="Green" title="E28">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E4:52_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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" title="ZFMISC_1:func.2">:]</a></span> &amp;  ex <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>] ) )
 <span class="kw">and </span><br/><a NAME="E5:52_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="partfun1.html#S2" title="PARTFUN1:sch.2">PARTFUN1:sch 2</a>(<a class="txt" href="#E1:52_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E2:52_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c1">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  ex <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] ) ) ) &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:52_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  ex <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> iff ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</font>] ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:52_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> implies ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</font>] ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</font>] implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:52_1_3_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</font>] )</span><br/>

<span class="kw">hence </span><a NAME="E2:52_1_3_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:52_1_3_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,<font color="Maroon" title="c4">z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E1:52_1_3_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>]</span><br/>

<span class="kw">thus </span><a NAME="E5:52_1_3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_3_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:52_1_3"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</font>] or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> )</span><br/>

<a NAME="E3:52_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c4">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E5:52_1_3"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,<font color="Maroon" title="c5">z</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">y1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c5">z</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:52_1_3_2"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c5">z</font>]</span><br/><a NAME="E2:52_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E3:52_1_3_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c5">z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_1_3"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E1:52_1_3_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:52_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E3:52_1_3"><span class="lab"><font color="Green" title="E28">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:52_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
