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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FX</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">FX</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" title="PCOMPS_2:func.1">PartUnion</a> (<font color="Olive" title="b1">B</font>,<font color="Maroon" title="c2">R</font>)</span>)</span> );<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:4_1_1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">FX</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xfamily.html#S1" onmouseover="rs('xfamily/S1')" onmouseout="rh()">XFAMILY:sch 1</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c3">X</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">X</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff  ex <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FX</font> st <br/>( <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">FX</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" title="PCOMPS_2:func.1">PartUnion</a> (<font color="Olive" title="b2">B</font>,<font color="Maroon" title="c2">R</font>)</span>)</span> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff  ex <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FX</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">FX</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" title="PCOMPS_2:func.1">PartUnion</a> (<font color="Olive" title="b1">B</font>,<font color="Maroon" title="c2">R</font>)</span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:4_1_1"/>
( <font color="Maroon" title="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies  ex <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FX</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">FX</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" title="PCOMPS_2:func.1">PartUnion</a> (<font color="Olive" title="b1">B</font>,<font color="Maroon" title="c2">R</font>)</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FX</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">FX</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" title="PCOMPS_2:func.1">PartUnion</a> (<font color="Olive" title="b1">B</font>,<font color="Maroon" title="c2">R</font>)</span>)</span> ) implies <font color="Maroon" title="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:4_1_1"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
 ex <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FX</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">FX</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" title="PCOMPS_2:func.1">PartUnion</a> (<font color="Olive" title="b1">B</font>,<font color="Maroon" title="c2">R</font>)</span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/>

<a NAME="E6:4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">FX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T109" onmouseover="rs('xboole_1/T109')" onmouseout="rh()">XBOOLE_1:109</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:4_1_1"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E5:4_1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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