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<b>let </b><font color="Maroon" title="c1">A</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"> for <font color="Olive" title="b1">a</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><br/> for <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" title="NORMFORM:func.12">NormForm</a> <font color="Maroon" title="c1">A</font></span>)</span>  st ( for <font color="Olive" title="b4">b</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 <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">u</font> holds <br/><font color="Olive" title="b4">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> ) &amp; ( for <font color="Olive" title="b4">c</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 <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">u</font> holds <br/> ex <font color="Olive" title="b5">b</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="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">v</font> &amp; <font color="Olive" title="b5">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Olive" title="b4">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Olive" title="b1">a</font> ) ) holds <br/> ex <font color="Olive" title="b4">b</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="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Olive" title="b2">u</font></span>)</span> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Olive" title="b3">v</font></span>)</span> &amp; <font color="Olive" title="b4">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Olive" title="b1">a</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font> be    <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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" title="NORMFORM:func.12">NormForm</a> <font color="Maroon" title="c1">A</font></span>)</span>  st ( for <font color="Olive" title="b3">b</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 <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> holds <br/><font color="Olive" title="b3">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> ) &amp; ( for <font color="Olive" title="b3">c</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 <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> holds <br/> ex <font color="Olive" title="b4">b</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="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">v</font> &amp; <font color="Olive" title="b4">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Olive" title="b3">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> ) ) holds <br/> ex <font color="Olive" title="b3">b</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="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Olive" title="b2">v</font></span>)</span> &amp; <font color="Olive" title="b3">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )</span><br/><b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" title="NORMFORM:func.12">NormForm</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">b</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 <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> holds <br/><font color="Olive" title="b1">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> ) &amp; ( for <font color="Olive" title="b1">c</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 <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> holds <br/> ex <font color="Olive" title="b2">b</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="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Olive" title="b2">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Olive" title="b1">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> ) ) implies  ex <font color="Olive" title="b1">b</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="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c4">v</font></span>)</span> &amp; <font color="Olive" title="b1">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> ) )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:32"/>
for <font color="Olive" title="b1">b</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 <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> holds <br/><font color="Olive" title="b1">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">b</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 <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> holds <br/> ex <font color="Olive" title="b2">c</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="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Olive" title="b2">c</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Olive" title="b1">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">b</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="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c4">v</font></span>)</span> &amp; <font color="Olive" title="b1">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>defpred </b><font color="Maroon">S<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#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>] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c4">v</font> &amp; $2 <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> $1 <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> );<br/>
<div><i><font color="Green" title="E28">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">b</font> be    <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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font> implies  ex <font color="Olive" title="b1">x</font> being   <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> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">b</font>,<font color="Olive" title="b1">x</font>] )</span><br/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <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> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">b</font>,<font color="Olive" title="b1">x</font>]</span><br/><b>then consider </b><font color="Maroon" title="c6">c</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><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c6">c</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="heyting1.html#E2:32"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">c</font> as    <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> ;<br/><b>take </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">x</font>]</span><br/><b>thus </b><a NAME="E5:32_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">x</font>]
 <b>by </b><i><a class="txt" href="heyting1.html#E3:32_1"><i><font color="Green" title="E29">A3</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><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">f'</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</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>,<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><b> such that </b><br/><a NAME="E5:32"/><i><font color="Green" title="E29">A4</font></i>: 
for <font color="Olive" title="b1">b</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 <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font>]
 <b>from </b><i><a class="ref" href="fraenkel.html#S27" title="FRAENKEL:sch.27">FRAENKEL:sch 27</a>(<a class="txt" href="heyting1.html#E3:32"><i><font color="Green" title="E28">A2</font></i></a>);<br/></i>
<a NAME="E6:32"/>
for <font color="Olive" title="b1">b</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 <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> holds <br/><font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font>
 
<b>by </b><i><a class="txt" href="heyting1.html#E5:32"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/>
<a NAME="E7:32"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c5">f'</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">v</font>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T15" title="SETWISEO:th.15">SETWISEO:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">d</font> =  <a href="normform.html#K6" title="NORMFORM:func.6">FinPairUnion</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c5">f'</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>;<br/>
<div><i><font color="Green" title="E31">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">s</font> be    <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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c5">f'</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )</span><br/><a NAME="E1:32_2"/><i><font color="Green" title="E32">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c5">f'</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="normform.html#K3" title="NORMFORM:func.3">\</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E36"><i><font color="Green" title="E25">Lm5</font></i></a></i>;<br/><b>assume </b><a NAME="E2:32_2"/>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c5">f'</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font></span><br/><a NAME="E3:32_2"/><b>then </b>
<font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E5:32"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:32_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c5">f'</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:32_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="normform.html#T31" title="NORMFORM:th.31">NORMFORM:31</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>
<a NAME="E9:32"/><b>then </b>
 <a href="normform.html#K6" title="NORMFORM:func.6">FinPairUnion</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c5">f'</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="normform.html#T40" title="NORMFORM:th.40">NORMFORM:40</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">d</font> =  <a href="normform.html#K6" title="NORMFORM:func.6">FinPairUnion</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c5">f'</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> as    <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> <b>by </b><i><a class="ref" href="normform.html#T45" title="NORMFORM:th.45">NORMFORM:45</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c4">v</font></span>)</span> &amp; <font color="Maroon" title="c7">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<a NAME="E11:32"/>
<font color="Maroon" title="c7">d</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="normform.html#K6" title="NORMFORM:func.6">FinPairUnion</a> <span class="p3">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Olive" title="b1">f</font>,<span class="p4">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p5">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</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>,<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> : <font color="Olive" title="b1">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">v</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="heyting1.html#E7:32"><i><font color="Green" title="E30">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:32"/>
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c4">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font></span><br/>

<b>thus </b><a NAME="E13:32"/>
<font color="Maroon" title="c7">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E8:32"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="normform.html#T40" title="NORMFORM:th.40">NORMFORM:40</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>
