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

<b>set </b><font color="Maroon" title="c1">Z</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c2">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> ;<br/>
<a NAME="E1:14_1"/><i><font color="Green" title="E11">A0</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:14_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c4">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1"/><i><font color="Green" title="E12">A3</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:14_1_1"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">z</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">z</font>] )
 <b>by </b><i><a class="txt" href="#E3:14_1_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>
<a NAME="E6:14_1_1"/>
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D1" target="_self" title="ABCMIZ_1:def.1">VARCL</a></i>;<br/>
<a NAME="E7:14_1_1"/><b>then </b>
(  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c4">Y</font> )
 
<b>by </b><i><a class="txt" href="#E3:14_1_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="#E5:14_1_1"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:14_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  implies <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  )</span><br/><b>assume </b><a NAME="E1:14_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> </span><br/><b>then consider </b><font color="Maroon" title="c5">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_1_2"/><i><font color="Green" title="E12">A1</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:14_1_2"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">z</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">z</font>] )
 <b>by </b><i><a class="txt" href="#E3:14_1_2"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/><a NAME="E6:14_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E3:14_1_2"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E5:14_1_2"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="abcmiz_1.html#D1" target="_self" title="ABCMIZ_1:def.1">VARCL</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E7:14_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1"/>
 <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:14_1"><i><font color="Green" title="E11">A0</font></i></a>, <a class="ref" href="abcmiz_1.html#D1" target="_self" title="ABCMIZ_1:def.1">VARCL</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> </span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E4:14_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> </span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c4">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:14_1"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>)</span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E8:14_1"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">z</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">z</font>] )
 <b>by </b><i><a class="txt" href="#E6:14_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>
<a NAME="E9:14_1"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">z</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E8:14_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<a NAME="E10:14_1"/><b>then </b>
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E8:14_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="abcmiz_1.html#T9" target="_self" title="ABCMIZ_1:th.9">Th13</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:14_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span> </span>)</span>
 <b>by </b><i><a class="txt" href="#E6:14_1"><i><font color="Green" title="E12">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
