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

<b>let </b><font color="Maroon">Q</font> be   <a href="quantal1.html#NM2">Quantale</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="quantal1.html#NM1">UnOp</a> of <font color="Maroon">Q</font>;<br/>



<b>assume </b><a NAME="E1:74"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">j</font> is <a href="quantal1.html#V14" target="_self">monotone</a> &amp; <font color="Maroon">j</font> is <a href="quantal1.html#V11" target="_self">idempotent</a> &amp; <font color="Maroon">j</font> is <a href="quantal1.html#V15" target="_self">\/-distributive</a> )
 ;<br/>

<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/>
<a NAME="E3:74"/><i><font color="Green">E27</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">Q</font>) &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">Q</font>) &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">Q</font>) )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">D</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">j</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Q</font> <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">Q</font>),<font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> = <font color="Maroon">j'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">Q</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font>,<font color="Maroon">D</font><b> such that </b><br/><a NAME="E7:74"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">X</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E8:74"/><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:74"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font>;<br/>

<b>consider </b><font color="Maroon">a'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:74_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">j</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> <b>by </b><i><a class="ref" href="quantal1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Q</font> ;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> = 
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="quantal1.html#T1" target="_self">Th1</a>, <a class="ref" href="lattice3.html#T43">LATTICE3:43</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="quantal1.html#T1" target="_self">Th1</a>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:74"/>
for <font color="Olive">X</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">D</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Olive">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">D</font>;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>, <font color="Maroon">X'</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Q</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E2:74_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:74_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E3:74_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> )
 ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:74_2_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="quantal1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Q</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:74_2_1"/>
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:74_2_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span> ,<font color="Maroon">Q</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:74_2_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:74_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E3:74_2_1_1"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E4:74_2_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:74_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:74_2_1"/><b>then </b>
<font color="Maroon">a'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>
<a NAME="E12:74_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="quantal1.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E13:74_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X'</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:74_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X'</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E15:74_2_1"/>
<font color="Maroon">a</font> <a href="lattices.html#R1">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:74_2"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/><b>assume </b><a NAME="E1:74_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E3:74_2_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> )
 ;<br/><b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:74_2_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">Q</font></span>)</span>;<br/><a NAME="E7:74_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><a NAME="E8:74_2_2"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> &amp; <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">Q</font> )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/><b>hence </b><a NAME="E9:74_2_2"/>
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> </span>}</span>  )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E5:74_2"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T48">LATTICE3:48</a></i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> = 
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="quantal1.html#T1" target="_self">Th1</a>, , <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>
<br/>.= 
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">L</font> being  <a href="lattices.html#V3">strict</a> <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a><b> such that </b><br/><a NAME="E12:74"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">L</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> holds   <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font> ) )
 <b>by </b><i>, <a class="ref" href="lattice3.html#T56">LATTICE3:56</a></i>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>thus </b><a NAME="E13:74"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">L</font>) <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="quantal1.html#T1" target="_self">Th1</a></i>;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>thus </b> <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="quantal1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">Q</font></span>)</span>
<b>by </b><i>, <a class="ref" href="quantal1.html#T1" target="_self">Th1</a></i>
;<br/>


</div>
