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

<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> : <font color="Olive">x</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> </span>}</span> ;<br/>
<a NAME="E1:6_1_1"/>
 <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> : <font color="Olive">x</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> : <font color="Olive">x</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">a<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>3</sub></font> &amp; ( for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">a<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">x'</font> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">x'</font> ) );<br/>
<a NAME="E3:6_1_1"/><i><font color="Green">E34</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">B</font>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>

<a NAME="E1:6_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
;<br/>
<a NAME="E2:6_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 ex <font color="Olive">x''</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x''</font> &amp; <font color="Olive">x''</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 
;<br/>
<a NAME="E3:6_1_1_1"/>
 ex <font color="Olive">y''</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y''</font> &amp; <font color="Olive">y''</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:6_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font> &amp; ( for <font color="Olive">x'</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>  st <font color="Maroon">x1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">y1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x'</font> holds <br/><font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x'</font> ) )
 <b>by </b><i><a class="ref" href="eqrel_1.html#T20">EQREL_1:20</a></i>;<br/>
<a NAME="E7:6_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<b>take </b>
<font color="Maroon">z'</font>
;<br/>

<b>thus </b><a NAME="E9:6_1_1_1"/>
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z'</font>
 <b>by </b><i/>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><b>assume </b><a NAME="E1:6_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font>
 ;<br/><a NAME="E2:6_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E3:6_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x''</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x''</font> &amp; <font color="Olive">x''</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x'</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> ;<br/><a NAME="E5:6_1_1_1_1"/>
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x1'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:6_1_1_1_1"/>
<font color="Maroon">z'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">B1</font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:6_1_1"/><i><font color="Green">E39</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">B</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:6_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/><b>reconsider </b><font color="Maroon">B1'</font> = <font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/><a NAME="E4:6_1_1_2"/>
<font color="Maroon">B1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E5:6_1_1_2"/><b>then </b>
 ex <font color="Olive">B1''</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">B1'</font> <a href="hidden.html#R1">=</a> <font color="Olive">B1''</font> &amp; <font color="Olive">B1''</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">B1'</font> = <font color="Maroon">B1'</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> ;<br/><a NAME="E7:6_1_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> &amp; ( for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>  st <font color="Maroon">x1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x'</font> holds <br/><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x'</font> ) )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x'</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:6_1_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font>
 ;<br/><a NAME="E2:6_1_1_2_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/><a NAME="E4:6_1_1_2_1"/>
<font color="Maroon">x</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x1'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:6_1_1_2_1"/>
<font color="Maroon">B1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E9:6_1_1_2"/><b>then </b>
<font color="Maroon">B1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/><b>hence </b><a NAME="E10:6_1_1_2"/>
<font color="Maroon">B1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>] <b>means </b><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E7:6_1_1"/><i><font color="Green">E43</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">B</font>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>

<a NAME="E1:6_1_1_3"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
;<br/>
<a NAME="E2:6_1_1_3"/><b>then </b><i><font color="Green">E45</font></i>: 
 ex <font color="Olive">x1</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 
;<br/>
<a NAME="E3:6_1_1_3"/>
 ex <font color="Olive">y1</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y1</font> &amp; <font color="Olive">y1</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 
<b>by </b><i/>;<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="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">x'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">y'</font>;<br/>
<a NAME="E5:6_1_1_3"/>
<font color="Maroon">x'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">y'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E7:6_1_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">B2</font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E9:6_1_1"/><i><font color="Green">E46</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">B</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:6_1_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</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="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/><a NAME="E3:6_1_1_4"/>
<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:6_1_1_4"/>
<font color="Maroon">B2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">B</font>,<font color="Maroon">B1</font>,<font color="Maroon">B2</font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a>  ;<br/>
<a NAME="E12:6_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">B1</font> is <a href="binop_1.html#V1">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><a NAME="E1:6_1_1_5_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E2:6_1_1_5_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 ex <font color="Olive">x1</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 ;<br/><a NAME="E3:6_1_1_5_1"/>
 ex <font color="Olive">x2</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp; <font color="Olive">x2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">a</font>, <font color="Maroon">U2</font> = <font color="Maroon">b</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">U2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">U1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_1_5"/>
<font color="Maroon">B1</font> is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:6_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">B1</font> is <a href="binop_1.html#V2">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><a NAME="E1:6_1_1_6_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E2:6_1_1_6_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 ex <font color="Olive">x1</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 ;<br/><a NAME="E3:6_1_1_6_1"/><i><font color="Green">E82</font></i>: 
 ex <font color="Olive">x2</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp; <font color="Olive">x2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E4:6_1_1_6_1"/>
 ex <font color="Olive">x3</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">x3</font> &amp; <font color="Olive">x3</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">a</font>, <font color="Maroon">U2</font> = <font color="Maroon">b</font>, <font color="Maroon">U3</font> = <font color="Maroon">c</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">B1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">U3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">U1</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">U3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">U2</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">U3</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">U2</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_1_6"/>
<font color="Maroon">B1</font> is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:6_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">B2</font> is <a href="binop_1.html#V1">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><a NAME="E1:6_1_1_7_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E2:6_1_1_7_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 ex <font color="Olive">x1</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 ;<br/><a NAME="E3:6_1_1_7_1"/>
 ex <font color="Olive">x2</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp; <font color="Olive">x2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">a</font>, <font color="Maroon">U2</font> = <font color="Maroon">b</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">U2</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">U1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_1_7"/>
<font color="Maroon">B2</font> is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:6_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">B2</font> is <a href="binop_1.html#V2">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><a NAME="E1:6_1_1_8_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E2:6_1_1_8_1"/><b>then </b><i><font color="Green">E90</font></i>: 
 ex <font color="Olive">x1</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 ;<br/><a NAME="E3:6_1_1_8_1"/><i><font color="Green">E91</font></i>: 
 ex <font color="Olive">x2</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp; <font color="Olive">x2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E4:6_1_1_8_1"/>
 ex <font color="Olive">x3</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">x3</font> &amp; <font color="Olive">x3</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">a</font>, <font color="Maroon">U2</font> = <font color="Maroon">b</font>, <font color="Maroon">U3</font> = <font color="Maroon">c</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">B2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">U3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">U1</font> <a href="eqrel_1.html#K4">/\</a> <span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">U3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">U2</font></span>)</span> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">U3</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">B2</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">U2</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_1_8"/>
<font color="Maroon">B2</font> is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:6_1_1"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_9"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
<font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E17:6_1_1"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_10"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">z</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E2:6_1_1_10"/><i><font color="Green">E94</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font></span>)</span> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E18:6_1_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_11"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E2:6_1_1_11"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_11_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_1_11_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
;<br/>
<a NAME="E2:6_1_1_11_1"/><b>then </b><i><font color="Green">E98</font></i>: 
 ex <font color="Olive">x1</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 
;<br/>
<a NAME="E3:6_1_1_11_1"/>
 ex <font color="Olive">x2</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp; <font color="Olive">x2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">x</font>, <font color="Maroon">U2</font> = <font color="Maroon">y</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E5:6_1_1_11_1"/>
<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">U2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">y</font> = 
<font color="Maroon">U2</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">U2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="eqrel_1.html#T25">EQREL_1:25</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E19:6_1_1"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_12"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> = 
<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E20:6_1_1"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_13"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">z</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E2:6_1_1_13"/><i><font color="Green">E101</font></i>: 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font></span>)</span> = 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E21:6_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_14"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E2:6_1_1_14"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_14_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_1_14_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
;<br/>
<a NAME="E2:6_1_1_14_1"/><b>then </b><i><font color="Green">E104</font></i>: 
 ex <font color="Olive">x1</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 
;<br/>
<a NAME="E3:6_1_1_14_1"/>
 ex <font color="Olive">x2</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp; <font color="Olive">x2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">x</font>, <font color="Maroon">U2</font> = <font color="Maroon">y</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E5:6_1_1_14_1"/>
<font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">U2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">U1</font> <a href="eqrel_1.html#K4">/\</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">U2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="eqrel_1.html#T24">EQREL_1:24</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E22:6_1_1"/><b>then </b>
( <font color="Maroon">L</font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V5">join-associative</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V8">meet-absorbing</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V7">meet-associative</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i>, , , , , <a class="ref" href="lattices.html#D4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6">LATTICES:def 6</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#D9">LATTICES:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as  <a href="lattices.html#V3">strict</a> <a href="lattices.html#NM1">Lattice</a> <b>by </b><i><a class="ref" href="lattices.html#D10">LATTICES:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>thus </b><a NAME="E24:6_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> : <font color="Olive">x</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E25:6_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> holds <br/> ( the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="eqrel_1.html#K4">/\</a> <font color="Olive">y</font> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">y</font> )
 <b>by </b><i>, </i>;<br/>


</div>
