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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font> holds   <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b1">K</font>,<span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Olive" title="b1">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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><br/>



<a NAME="E1:20"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
( the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T27" title="LATTICE2:th.27">LATTICE2:27</a>, <a class="ref" href="lattice2.html#T29" title="LATTICE2:th.29">LATTICE2:29</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <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>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font> =  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> $1;<br/>
<a NAME="E2:20"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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> is  <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T26" title="LATTICE2:th.26">LATTICE2:26</a></span>;<br/>
<a NAME="E3:20"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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> <a href="heyting1.html#R1" title="HEYTING1:pred.1">c=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="heyting1.html#D1" target="_self" title="HEYTING1:def.1">HEYTING1:def 1</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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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> implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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>
 ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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><span class="kw"> such that </span><br/><a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E4:20_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#T70" title="SETWISEO:th.70">SETWISEO:70</a></span>;<br/>
<a NAME="E5:20_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="setwiseo.html#T68" title="SETWISEO:th.68">SETWISEO:68</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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> &amp; <font color="Maroon" title="c5">s</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c3">a</font> implies <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_1_1"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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 class="kw">and </span><br/><a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c3">a</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="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E4:20_1_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E5:20_1_1"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="setwiseo.html#T70" title="SETWISEO:th.70">SETWISEO:70</a></span>;<br/><a NAME="E6:20_1_1"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="setwiseo.html#T68" title="SETWISEO:th.68">SETWISEO:68</a></span>;<br/><span class="kw">hence </span><a NAME="E7:20_1_1"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="normform.html#T52" title="NORMFORM:th.52">NORMFORM:52</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="E7:20_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="normform.html#T61" title="NORMFORM:th.61">NORMFORM:61</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="E4:20"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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="heyting1.html#R1" title="HEYTING1:pred.1">c=</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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 class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T64" title="NORMFORM:th.64">NORMFORM:64</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p2">(<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 class="p1">(<span class="default"><a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:20"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
for <font color="Olive" title="b1">B</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="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p2">(<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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#D14" title="NORMFORM:def.14">NORMFORM:def 14</a></span>;<br/>
<span class="lab"><font color="Green" title="E16">A14</font></span>: <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p2">(<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> = 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p2">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E15">A13</font></span></a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T64" title="NORMFORM:th.64">NORMFORM:64</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T86" title="NORMFORM:th.86">NORMFORM:86</a></span>
<br/>.= 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T28" title="LATTICE2:th.28">LATTICE2:28</a>, <a class="ref" href="lattice2.html#T67" title="LATTICE2:th.67">LATTICE2:67</a></span>
;<br/>
<div><span class="lab"><font color="Green" title="E17">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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 href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><br/><a NAME="E1:20_3"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
(  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E15">A13</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c6">y</font></span>)</span> = 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E15">A13</font></span></a></span>
<br/>.= 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T68" title="NORMFORM:th.68">NORMFORM:68</a></span>
<br/>.= 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T69" title="NORMFORM:th.69">NORMFORM:69</a></span>
<br/>.= 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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 href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_3"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="normform.html#D14" title="NORMFORM:def.14">NORMFORM:def 14</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>
<div><span class="lab"><font color="Green" title="E18">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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="c5">a</font> = 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="heyting1.html#K2" title="HEYTING1:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="heyting1.html#K2" title="HEYTING1:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#T67" title="SETWISEO:th.67">SETWISEO:67</a></span>
<br/>.= 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1"><a href="heyting1.html#K2" title="HEYTING1:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="heyting1.html#K2" title="HEYTING1:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E15">A13</font></span></a></span>
<br/>.= 
<span class="p1"><a href="heyting1.html#K2" title="HEYTING1:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="heyting1.html#K2" title="HEYTING1:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T66" title="NORMFORM:th.66">NORMFORM:66</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="heyting1.html#D4" target="_self" title="HEYTING1:def.4">Def4</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 href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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 href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#D3" title="LATTICE2:def.3">LATTICE2:def 3</a></span>
<br/>.= 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" title="NORMFORM:func.12">NormForm</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="lattice2.html#T67" title="LATTICE2:th.67">LATTICE2:67</a>, <a class="ref" href="setwiseo.html#T65" title="SETWISEO:th.65">SETWISEO:65</a></span>
<br/>.= 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E15">A13</font></span></a></span>
<br/>.= 
 <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <span class="p2">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
