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<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span> holds  <font color="Olive" title="b1">B</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b1">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</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="c2">B</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c2">B</font></span><br/>





<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font>) <span class="kw">-&gt; </span>   <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"><span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = $1 <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> $2;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>( $1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; $2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>( $2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; $1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">B</font> );<br/>
<span class="kw">set </span><font color="Maroon" title="c4">X1</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">X2</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span> } </span> ;<br/>
<a NAME="E1:49"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>)
 
;<br/>
<div><span class="lab"><font color="Green" title="E39">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">t</font>,<font color="Olive" title="b2">s</font>) where <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">s</font>,<font color="Olive" title="b3">t</font>] </span> } </span>  iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">t</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Olive" title="b3">s</font></span>)</span> where <font color="Olive" title="b2">t</font>, <font color="Olive" title="b3">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b3">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span>  )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span> } </span>  iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span>  )</span><br/><a NAME="E1:49_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span> } </span>  iff  ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) )
 ;<br/><a NAME="E2:49_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span> } </span>  iff  ex <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) )
 ;<br/><span class="kw">hence </span><a NAME="E3:49_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span> } </span>  iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">B</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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:49"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] )
 
;<br/>
<a NAME="E4:49"/>
<span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span> } </span> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S8" onmouseover="rs('fraenkel/S8')" onmouseout="rh()">FRAENKEL:sch 8</a>(<a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E38">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:49"/>
<font color="Maroon" title="c2">B</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</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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