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

<b>let </b><font color="Maroon">B</font> be   <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>



<b>set </b><font color="Maroon">F</font> = <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span>;<br/>
<b>set </b><font color="Maroon">E</font> =  <a href="filter_0.html#K12">equivalence_wrt</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span>;<br/>
<a NAME="E1:84"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">B</font> is   <a href="lattices.html#NM4">0_Lattice</a> &amp; <font color="Maroon">B</font> is   <a href="lattices.html#NM5">1_Lattice</a> &amp; <font color="Maroon">B</font> is   <a href="filter_0.html#NM2">I_Lattice</a> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T40">FILTER_0:40</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:84"/><i><font color="Green">E69</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E4:84"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> holds  <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>

<a NAME="E1:84_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K7">`</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K7">`</a>  &amp; <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> &amp; <font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> &amp; <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</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#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="lattices.html#T18">LATTICES:18</a>, <a class="ref" href="lattices.html#T39">LATTICES:39</a>, <a class="ref" href="lattices.html#T40">LATTICES:40</a>, <a class="ref" href="lattices.html#T47">LATTICES:47</a>, <a class="ref" href="lattices.html#T51">LATTICES:51</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">b</font> = 
<font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">B</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T48">LATTICES:48</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattices.html#T43">LATTICES:43</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:84"/><i><font color="Green">E78</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span>
 
<b>by </b><i><a class="ref" href="filter_0.html#T63">FILTER_0:63</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="binop_1.html#NM1">UnOp</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E7:84"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</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>
<b>take </b><font color="Maroon">f</font> = <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K13">:&gt;</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a>,<a class="ref" href="filter_1.html#D9" target="_self">FILTER_1:def 9</a><br/>

<b>set </b><font color="Maroon">R</font> =  <a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span>;<br/>
<a NAME="E8:84"/><i><font color="Green">E80</font></i>: 
(  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p2"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:84"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:84"/><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_3.html#T70">FUNCT_3:70</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">B</font>), <font color="Maroon">o2</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">B</font>) as   <a href="filter_1.html#NM4">BinOp</a> of  <a href="filter_0.html#K12">equivalence_wrt</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> <b>by </b><i>, , </i>;<br/>
<a NAME="E12:84"/><i><font color="Green">E86</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> &amp;  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="filter_1.html#K3" target="_self">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="filter_1.html#K3" target="_self">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span>)</span> #) <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> )
 
<b>by </b><i>, , <a class="ref" href="filter_0.html#T63">FILTER_0:63</a></i>;<br/>
<a NAME="E13:84"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>

<a NAME="E1:84_2"/>
<font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:84_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">b</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_0.html#K12">equivalence_wrt</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E3:84_2"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> )
 
<b>by </b><i>, <a class="ref" href="filter_0.html#D12">FILTER_0:def 12</a></i>;<br/>
<b>hence </b><a NAME="E4:84_2"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="filter_0.html#T63">FILTER_0:63</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E14:84"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p2"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:84_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_3"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_3.html#T70">FUNCT_3:70</a></i>;<br/>
<a NAME="E5:84_3"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K10">EqClass</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span> <b>by </b><i><a class="ref" href="filter_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:84_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:84_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p2"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E15:84"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p2"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:84_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p2"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p4"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>, <font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span><b> such that </b><br/><a NAME="E3:84_4"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:84_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<b>set </b><font color="Maroon">ty</font> = <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E6:84_4"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, </i>;<br/>
<a NAME="E7:84_4"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E8:84_4"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:84_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
;<br/>
<a NAME="E10:84_4"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_0.html#K12">equivalence_wrt</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span>
 
<b>by </b><i><a class="ref" href="filter_0.html#D12">FILTER_0:def 12</a></i>;<br/>
<a NAME="E11:84_4"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E12:84_4"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E13:84_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:84_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, <a class="ref" href="filter_1.html#D1" target="_self">Def1</a>, , </i>;<br/>
<a NAME="E15:84_4"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, <a class="ref" href="filter_1.html#D2" target="_self">Def2</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>
<a NAME="E16:84_4"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:84_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E18:84_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E16:84"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:84_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="funct_3.html#T70">FUNCT_3:70</a></i>;<br/>
<a NAME="E3:84_5"/><i><font color="Green">E204</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">x'</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y'</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">x'</font> &amp; <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y'</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">y'</font> )
 
<b>by </b><i>, , , , , <a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>;<br/>
<b>assume </b><a NAME="E4:84_5"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E5:84_5"/><b>then </b><i><font color="Green">E205</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E6:84_5"/><b>then </b>
( <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, </i>;<br/>
<a NAME="E7:84_5"/><b>then </b>
<font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>
<b>hence </b><a NAME="E8:84_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, <a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, </i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E17:84"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_6"/><i><font color="Green">E206</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font>
 ;<br/>

<b>hence </b><a NAME="E2:84_6"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i>, <a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:84_6"/><i><font color="Green">E207</font></i>: 
( <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="lattices.html#T43">LATTICES:43</a></i>;<br/>
<a NAME="E5:84_6"/><b>then </b>
(  <a href="lattices.html#K6">Top</a> <font color="Maroon">B</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">y'</font> &amp;  <a href="lattices.html#K6">Top</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> )
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T12">FILTER_0:12</a>, <a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a></i>;<br/>
<a NAME="E6:84_6"/><b>then </b>
<font color="Maroon">x'</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span>
 
<b>by </b><i><a class="ref" href="filter_0.html#T9">FILTER_0:9</a></i>;<br/>
<a NAME="E7:84_6"/><b>then </b><i><font color="Green">E208</font></i>: 
( <font color="Maroon">x'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> &amp; <font color="Maroon">x'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">x'</font> &amp;  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">y'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y'</font> &amp;  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E8:84_6"/><i><font color="Green">E209</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , , , <a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, <a class="ref" href="filter_0.html#T1">FILTER_0:1</a>, <a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>;<br/>
<a NAME="E9:84_6"/>
( <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">x'</font> &amp; <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y'</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">hx</font> = <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>, <font color="Maroon">hy</font> = <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span> <b>by </b><i>, , , </i>;<br/>
<a NAME="E11:84_6"/>
<font color="Maroon">hx</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">hy</font>
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T65">FILTER_0:65</a></i>;<br/>
<a NAME="E12:84_6"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">hx</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">hy</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i><a class="ref" href="filter_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E13:84_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="filter_1.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E18:84"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font></span>)</span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<a NAME="E20:84"/><i><font color="Green">E212</font></i>: 
( <font color="Maroon">x'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">x'</font> &amp; <font color="Maroon">y'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y'</font> &amp;  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> &amp;  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , , , , , , <a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>;<br/>
<a NAME="E21:84"/>
( <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">x'</font> &amp; <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">y'</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">hx</font> = <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>, <font color="Maroon">hy</font> = <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p2"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span> <b>by </b><i>, , <a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, </i>;<br/>
<b>assume </b><a NAME="E23:84"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8" target="_self">LattRel</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="filter_1.html#K4" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="filter_0.html#K8">latt</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span></span><a href="filter_1.html#K7" target="_self">:]</a></span>
 ;<br/>

<a NAME="E24:84"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">hx</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">hy</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, </i>;<br/>
<a NAME="E25:84"/><b>then </b>
( <font color="Maroon">x'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font> <a href="filter_1.html#K5" target="_self">/\/</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> &amp; <font color="Maroon">hx</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">hy</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E26:84"/><b>then </b>
( <font color="Maroon">x'</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">a</font></span><a href="filter_0.html#K2">.)</a></span> &amp; <font color="Maroon">x'</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="txt" href="filter_1.html#E1:84"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="filter_0.html#T55">FILTER_0:55</a>, <a class="ref" href="filter_0.html#T65">FILTER_0:65</a>, <a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E27:84"/><b>then </b>
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T18">FILTER_0:18</a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a>, <a class="ref" href="lattices.html#T27">LATTICES:27</a></i>;<br/>
<a NAME="E28:84"/><b>then </b><i><font color="Green">E213</font></i>: 
( <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y'</font></span>)</span> &amp; <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y'</font></span>)</span> &amp; <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y'</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">y'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y'</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a>, <a class="ref" href="lattices.html#T27">LATTICES:27</a>, <a class="ref" href="lattices.html#T39">LATTICES:39</a>, <a class="ref" href="lattices.html#T47">LATTICES:47</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E29:84"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<a NAME="E30:84"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y'</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, <a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<a NAME="E31:84"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:84"><i><font color="Green">E69</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E32:84"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8" target="_self">LattRel</a> <font color="Maroon">B</font>
 ;<br/>


</div>
