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

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/>





<a NAME="E1:37"/><i><font color="Green">E21</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i><a class="ref" href="lattice2.html#T27">LATTICE2:27</a></i>;<br/>
<a NAME="E2:37"/><i><font color="Green">E24</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> is <a href="binop_1.html#V2">associative</a>
 
<b>by </b><i><a class="ref" href="lattice2.html#T29">LATTICE2:29</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green">E25</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> is <a href="binop_1.html#V3">idempotent</a>
 
<b>by </b><i><a class="ref" href="lattice2.html#T26">LATTICE2:26</a></i>;<br/>
<a NAME="E4:37"/><i><font color="Green">E44</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 
<b>by </b><i><a class="ref" href="lattice2.html#T67">LATTICE2:67</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> =  <a href="substlat.html#K3">mi</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>, <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E6:37"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">B</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E7:37"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> ;<br/>
<i><font color="Green">E49</font></i>: <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> = 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="substlat.html#T9">SUBSTLAT:9</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="substlat.html#T26">SUBSTLAT:26</a></i>
<br/>.= 
 <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lattice2.html#T28">LATTICE2:28</a></i>
;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>;<br/><a NAME="E1:37_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="substlat.html#K3">mi</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E2:37_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="substlat.html#K3">mi</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">y</font></span>)</span> = 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substlat.html#K3">mi</a> <font color="Maroon">x</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="substlat.html#T13">SUBSTLAT:13</a></i>
<br/>.= 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substlat.html#K3">mi</a> <font color="Maroon">x</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><a href="substlat.html#K3">mi</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="substlat.html#T13">SUBSTLAT:13</a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="heyting2.html#T4" target="_self">Th4</a>, <a class="ref" href="heyting2.html#T5" target="_self">Th5</a>, <a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E11:37"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E12:37"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:37"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:37"/><i><font color="Green">E57</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
 
;<br/>
<a NAME="E15:37"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:37"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="heyting2.html#T6" target="_self">Th6</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E17:37"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p4">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> )
 
<b>by </b><i><a class="ref" href="heyting2.html#T4" target="_self">Th4</a>, <a class="ref" href="heyting2.html#T5" target="_self">Th5</a>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> <b>by </b><i><a class="ref" href="setwiseo.html#T14">SETWISEO:14</a></i>;<br/><a NAME="E3:37_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a'</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="heyting2.html#T8" target="_self">Th8</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">KL</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a'</font></span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p4">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="heyting2.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="setwiseo.html#T67">SETWISEO:67</a></i>
<br/>.= 
 <a href="substlat.html#K3">mi</a> <font color="Maroon">KL</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">KL</font>
<b>by </b><i><a class="ref" href="substlat.html#T11">SUBSTLAT:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="heyting2.html#T9" target="_self">Th9</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="heyting2.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E19:37"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E20:37"/><i><font color="Green">E91</font></i>: 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="heyting1.html#R1">c=</a>  <a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="substlat.html#T8">SUBSTLAT:8</a></i>;<br/>
<a NAME="E21:37"/><i><font color="Green">E92</font></i>: 
 <a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="heyting1.html#R1">c=</a>  <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:37_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E4:37_4"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T70">SETWISEO:70</a></i>;<br/>
<a NAME="E5:37_4"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="setwiseo.html#T68">SETWISEO:68</a></i>;<br/>
<a NAME="E6:37_4"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">a</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="heyting2.html#E6:37"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:37_4_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:37_4_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/><b>consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:37_4_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E5:37_4_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E10:37"><i><font color="Green">E50</font></i></a>, <a class="ref" href="setwiseo.html#T70">SETWISEO:70</a></i>;<br/><a NAME="E6:37_4_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E11:37"><i><font color="Green">E53</font></i></a>, <a class="ref" href="setwiseo.html#T68">SETWISEO:68</a></i>;<br/><a NAME="E7:37_4_1"/>
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 ;<br/><a NAME="E8:37_4_1"/><b>then </b>
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> : ( ( for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">u</font> is <a href="finset_1.html#V1">finite</a> ) &amp; ( for <font color="Olive">s</font>, <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substlat.html#D1">SUBSTLAT:def 1</a></i>;<br/><a NAME="E9:37_4_1"/><b>then </b>
 ex <font color="Olive">AA</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> st <br/>( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Olive">AA</font> &amp; ( for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">AA</font> holds <br/><font color="Olive">u</font> is <a href="finset_1.html#V1">finite</a> ) &amp; ( for <font color="Olive">s</font>, <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">AA</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">AA</font> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) )
 ;<br/><b>hence </b><a NAME="E10:37_4_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E6:37"><i><font color="Green">E46</font></i></a>, , , , <a class="txt" href="heyting2.html#E12:37"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:37_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="heyting2.html#E9:37"><i><font color="Green">E49</font></i></a>, <a class="ref" href="substlat.html#T7">SUBSTLAT:7</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">gs</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/>
<b>thus </b> <a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> = 
 <a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">K</font>,<font color="Maroon">gs</font>
<b>by </b><i><a class="ref" href="heyting2.html#T8" target="_self">Th8</a>, <a class="ref" href="lattice2.html#T69">LATTICE2:69</a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">K</font>,<font color="Maroon">gs</font>
<b>by </b><i><a class="ref" href="lattice2.html#D3">LATTICE2:def 3</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="heyting2.html#T3" target="_self">Th3</a>, , , <a class="txt" href="heyting2.html#E4"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="setwiseo.html#T65">SETWISEO:65</a></i>
<br/>.= 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p3">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="setwiseo.html#K10">FinUnion</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11">singleton</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="heyting2.html#T9" target="_self">Th9</a>, <a class="txt" href="heyting2.html#E4:37"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/>


</div>
