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





<b>set </b><font color="Maroon">A</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">d</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">c</font>;<br/>
<a NAME="E1:83"/><i><font color="Green">E68</font></i>: 
<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/>
<a NAME="E2:83"/>
( <font color="Maroon">d</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">d</font></span><a href="filter_0.html#K2">.)</a></span> &amp; <font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">d</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:83"/><b>then </b>
( <font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">d</font> <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">d</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <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">d</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">c</font> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<b>hence </b><a NAME="E4:83"/>
<font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">d</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">d</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/>

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

<b>assume </b><a NAME="E5:83"/>
<font color="Maroon">b</font> <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">d</font></span><a href="filter_0.html#K2">.)</a></span></span>)</span>,<font color="Maroon">c</font>
 ;<br/>

<a NAME="E6:83"/><b>then </b>
<font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">c</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">d</font></span><a href="filter_0.html#K2">.)</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:83"/><b>then </b>
( <font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">c</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">c</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">c</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T18">FILTER_0:18</a></i>;<br/>
<a NAME="E8:83"/><b>then </b>
<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">c</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">c</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="lattices.html#K7">`</a> 
 
<b>by </b><i><a class="ref" href="lattices.html#T53">LATTICES:53</a></i>;<br/>
<a NAME="E9:83"/><b>then </b>
( <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">c</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">c</font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</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#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> &amp; <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">c</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">c</font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </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#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</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"><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">c</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</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#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <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> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> &amp; <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">c</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </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> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> &amp;  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span> &amp; <font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <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> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</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"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</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#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#T40">LATTICES:40</a>, <a class="ref" href="lattices.html#T47">LATTICES:47</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E10:83"/><b>then </b>
( <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">c</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">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">d</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">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font></span>)</span> &amp; <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">c</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">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</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">b</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">d</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">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">d</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</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">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T1">FILTER_0:1</a>, <a class="ref" href="lattices.html#T43">LATTICES:43</a>, <a class="ref" href="lattices.html#T48">LATTICES:48</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</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><a NAME="E11:83"/>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>


</div>
