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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> ;<br/><b>let </b><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>;<br/>



<a NAME="E1:16"/><i><font color="Green" title="E14">A1</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="lattice2.html#T27" title="LATTICE2:th.27">LATTICE2:27</a></i>;<br/>
<a NAME="E2:16"/><i><font color="Green" title="E15">A2</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="lattice2.html#T29" title="LATTICE2:th.29">LATTICE2:29</a></i>;<br/>
<a NAME="E3:16"/><i><font color="Green" title="E16">A3</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="lattice2.html#T26" title="LATTICE2:th.26">LATTICE2:26</a></i>;<br/>
<a NAME="E4:16"/><i><font color="Green" title="E17">A4</font></i>: 
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#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 
<b>by </b><i><a class="ref" href="lattice2.html#T67" title="LATTICE2:th.67">LATTICE2:67</a></i>;<br/>
<b>deffunc </b><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>) <b>-&gt; </b>   <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/>
<b>consider </b><font color="Maroon" title="c3">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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 href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E6:16"/><i><font color="Green" title="E18">A5</font></i>: 
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>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E7:16"/>
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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="normform.html#D14" title="NORMFORM:def.14">NORMFORM:def 14</a></i>;<br/>
<b>then reconsider </b><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  <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>,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> ;<br/>
<a NAME="E9:16"/><i><font color="Green" title="E19">A6</font></i>: 
(  <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> <a href="heyting1.html#R1" title="HEYTING1:pred.1">c=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</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> &amp;  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="normform.html#T64" title="NORMFORM:th.64">NORMFORM:64</a></i>;<br/>
<i><font color="Green" title="E20">A7</font></i>: <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>
<b>by </b><i><a class="txt" href="heyting1.html#E6:16"><i><font color="Green" title="E18">A5</font></i></a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="heyting1.html#E9:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>
<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>
<b>by </b><i><a class="ref" href="normform.html#T86" title="NORMFORM:th.86">NORMFORM:86</a></i>
<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>
<b>by </b><i><a class="txt" href="heyting1.html#E4:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="lattice2.html#T28" title="LATTICE2:th.28">LATTICE2:28</a></i>
;<br/>
<div><i><font color="Green" title="E21">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>let </b><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>;<br/><a NAME="E1:16_2"/><i><font color="Green" title="E22">A9</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="heyting1.html#E6:16"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/><a NAME="E2:16_2"/><i><font color="Green" title="E23">A10</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="heyting1.html#E6:16"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/><b>thus </b><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>
<b>by </b><i><a class="txt" href="heyting1.html#E6:16"><i><font color="Green" title="E18">A5</font></i></a></i>
<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>
<b>by </b><i><a class="ref" href="normform.html#T68" title="NORMFORM:th.68">NORMFORM:68</a></i>
<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>
<b>by </b><i><a class="ref" href="normform.html#T69" title="NORMFORM:th.69">NORMFORM:69</a></i>
<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#K2" title="BINOP_1:func.2">.</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>
<b>by </b><i><a class="txt" href="heyting1.html#E1:16_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="heyting1.html#E2:16_2"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="heyting1.html#E9"><i><font color="Green" title="E6">Lm3</font></i></a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E22">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>now </b></a><div class="add"><b>let </b><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>;<br/><b>thus </b><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>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<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>
<b>by </b><i><a class="ref" href="setwiseo.html#T67" title="SETWISEO:th.67">SETWISEO:67</a></i>
<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>
<b>by </b><i><a class="txt" href="heyting1.html#E6:16"><i><font color="Green" title="E18">A5</font></i></a></i>
<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>
<b>by </b><i><a class="ref" href="normform.html#T66" title="NORMFORM:th.66">NORMFORM:66</a></i>
<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>
<b>by </b><i><a class="ref" href="heyting1.html#D4" target="_self" title="HEYTING1:def.4">Def4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E13:16"/><i><font color="Green" title="E23">A12</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="normform.html#T64" title="NORMFORM:th.64">NORMFORM:64</a></i>;<br/>
<a NAME="E14:16"/><i><font color="Green" title="E24">A13</font></i>: 
 <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="16_4"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="heyting1.html#D1" target="_self" title="HEYTING1:def.1">HEYTING1:def 1</a><br/>

<b>assume </b><a NAME="E1:16_4"/><i><font color="Green" title="E25">A14</font></i>: 
<font color="Maroon" title="c5">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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">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><b> such that </b><br/><a NAME="E3:16_4"/><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <b>and </b><br/><a NAME="E4:16_4"/><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c5">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="c6">b</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T70" title="SETWISEO:th.70">SETWISEO:70</a></i>;<br/>
<a NAME="E5:16_4"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="heyting1.html#E4:16_4"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="setwiseo.html#T68" title="SETWISEO:th.68">SETWISEO:68</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4_1"><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>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:16_4_1"/><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c7">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>
 <b>and </b><br/><a NAME="E2:16_4_1"/><i><font color="Green" title="E30">A19</font></i>: 
<font color="Maroon" title="c7">s</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">a</font>
 ;<br/><b>consider </b><font color="Maroon" title="c8">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><b> such that </b><br/><a NAME="E4:16_4_1"/><i><font color="Green" title="E31">A20</font></i>: 
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <b>and </b><br/><a NAME="E5:16_4_1"/><i><font color="Green" title="E32">A21</font></i>: 
<font color="Maroon" title="c7">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="c8">t</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:16_4_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="setwiseo.html#T70" title="SETWISEO:th.70">SETWISEO:70</a></i>;<br/><a NAME="E6:16_4_1"/>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E5:16_4_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="setwiseo.html#T68" title="SETWISEO:th.68">SETWISEO:68</a></i>;<br/><b>hence </b><a NAME="E7:16_4_1"/>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E3:16_4"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="heyting1.html#E5:16_4"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="heyting1.html#E2:16_4_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="heyting1.html#E4:16_4_1"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="normform.html#T52" title="NORMFORM:th.52">NORMFORM:52</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:16_4"/>
<font color="Maroon" title="c5">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>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:16_4"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="normform.html#T61" title="NORMFORM:th.61">NORMFORM:61</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b> <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">$$</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice2.html#D3" title="LATTICE2:def.3">LATTICE2:def 3</a></i>
<br/>.= 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" title="NORMFORM:func.12">NormForm</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</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>
<b>by </b><i><a class="txt" href="heyting1.html#E12:16"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
<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>
<b>by </b><i><a class="txt" href="heyting1.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="heyting1.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="heyting1.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="heyting1.html#E4:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="heyting1.html#E10:16"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="heyting1.html#E11:16"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="setwiseo.html#T65" title="SETWISEO:th.65">SETWISEO:65</a></i>
<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>
<b>by </b><i><a class="txt" href="heyting1.html#E6:16"><i><font color="Green" title="E18">A5</font></i></a></i>
<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>
<b>by </b><i><a class="txt" href="heyting1.html#E13:16"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="heyting1.html#E14:16"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
;<br/>


</div>
