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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">A</font></span>)</span>] <b>means </b> ex <font color="Olive">E</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> iff <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> ) ) );<br/>
<a NAME="E1:33_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

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

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>] <b>means </b><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font>;<br/>
<b>consider </b><font color="Maroon">E</font> being    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">A</font>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:33_1_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2">RELSET_1:sch 2</a>();<br/></i>
<a NAME="E3:33_1_1_1"/>
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> <font color="Maroon">A</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E3:33_1_1_1_1"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<a NAME="E4:33_1_1_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:33_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:33_1_1_1"/><b>then </b>
<font color="Maroon">E</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E5:33_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<a NAME="E6:33_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_2"/><i><font color="Green">E47</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> <font color="Maroon">E</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E5:33_1_1_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:33_1_1_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:33_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:33_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">E</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<a NAME="E8:33_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_3"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_3"/><i><font color="Green">E51</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> <font color="Maroon">E</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E4:33_1_1_1_3"/>
( <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:33_1_1_1_3"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E6:33_1_1_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:33_1_1_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E8:33_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:33_1_1_1"/><b>then </b>
<font color="Maroon">E</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">E</font> = <font color="Maroon">E</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">E</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E13:33_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 ;<br/>
<b>thus </b><a NAME="E14:33_1_1_1"/>
 ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Olive">r</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S10">FUNCT_2:sch 10</a>();<br/></i>
<b>thus </b><a NAME="E4:33_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> st <br/>for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">E</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">E</font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> iff <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">e</font> ) ) )
 <b>by </b><i/>;<br/>


</div>
