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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>consider </b><font color="Maroon">M2</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E2:20_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:20_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E5:20_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">EqR</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> holds  <font color="Olive">EqR</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">EqR</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>

<i><font color="Green">E37</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">EqR</font> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E2:20_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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font> holds <br/><font color="Maroon">EqR</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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> ;<br/>

<b>assume </b><a NAME="E1:20_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</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">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:20_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:20_1_1_1_2"/>
<font color="Maroon">EqR</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">x'</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
;<br/>
<b>hence </b><a NAME="E5:20_1_1_1_2"/>
<font color="Maroon">EqR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_1"/>
<font color="Maroon">EqR</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>
<a NAME="E7:20_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<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>for <font color="Olive">x1</font>, <font color="Olive">y1</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>  st <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/> ex <font color="Olive">EqR3</font> being  <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">EqR3</font> <a href="pboole.html#R6">=</a> <font color="Olive">x1</font> <a href="pboole.html#K2">\/</a> <font color="Olive">y1</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_5.html#K3" target="_self">EqCl</a> <font color="Olive">EqR3</font> );<br/>
<a NAME="E9:20_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>  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="20_1_1_2"><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/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">x'</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">y'</font>;<br/>
<a NAME="E2:20_1_1_2"/>
<font color="Maroon">x'</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x'</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">y'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>let </b><font color="Maroon">x1</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>, <font color="Maroon">y1</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E4:20_1_1_2"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/>

<b>hence </b><a NAME="E5:20_1_1_2"/>
 ex <font color="Olive">EqR3</font> being  <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">EqR3</font> <a href="pboole.html#R6">=</a> <font color="Maroon">x1</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="msualg_5.html#K3" target="_self">EqCl</a> <font color="Olive">EqR3</font> )
 <b>by </b><i/>;<br/>


</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="E11:20_1_1"/><i><font color="Green">E40</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">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">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>, <font color="Maroon">y</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/><a NAME="E1:20_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</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="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">EqR3</font> being   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:20_1_1_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">EqR3</font> <a href="pboole.html#R6">=</a> <font color="Maroon">x</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">y</font> &amp; <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>  <a href="msualg_5.html#K3" target="_self">EqCl</a> <font color="Maroon">EqR3</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:20_1_1_3"/>
<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="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>3</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>for <font color="Olive">x1</font>, <font color="Olive">y1</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>  st <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> <a href="pboole.html#K3">/\</a> <font color="Olive">y1</font>;<br/>
<a NAME="E13:20_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>3</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="20_1_1_4"><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/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">x'</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">y'</font>;<br/>
<a NAME="E2:20_1_1_4"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">y'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1_1_4_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E3:20_1_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">y'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">y'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x'</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">y'</font> as   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="msualg_4.html#D2">MSUALG_4:def 2</a></i>;<br/>
<a NAME="E4:20_1_1_4"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">R</font> being  <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> holds <br/><font color="Olive">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>



<b>assume </b><a NAME="E1:20_1_1_4_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<b>reconsider </b><font color="Maroon">x1''</font> = <font color="Maroon">x'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font>, <font color="Maroon">y1''</font> = <font color="Maroon">y'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> ;<br/>
<b>reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x1''</font>, <font color="Maroon">y''</font> = <font color="Maroon">y1''</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/>
<a NAME="E5:20_1_1_4_2"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">y''</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1_1_4_2"/>
<font color="Maroon">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/>
<a NAME="E6:20_1_1_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font>
 
<b>by </b><i/>;<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> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E8:20_1_1_4"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</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="E15:20_1_1"/><i><font color="Green">E45</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>3</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">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>, <font color="Maroon">y</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/><a NAME="E1:20_1_1_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</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="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/><a NAME="E3:20_1_1_5"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/><b>hence </b><a NAME="E4:20_1_1_5"/>
<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="pboole.html#K3">/\</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<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="E18:20_1_1"/><i><font color="Green">E68</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="20_1_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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>;<br/><b>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="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</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">U1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">U2</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="E2:20_1_1_6"/>
<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="E19:20_1_1"/><i><font color="Green">E70</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="20_1_1_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><b>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="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</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>,<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="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">U3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">U1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="msualg_5.html#K4" target="_self">"\/"</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="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">U2</font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</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="msualg_5.html#K4" target="_self">"\/"</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="E2:20_1_1_7"/>
<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="E20:20_1_1"/><i><font color="Green">E73</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="20_1_1_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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>;<br/><b>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="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</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="pboole.html#K3">/\</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="E2:20_1_1_8"/>
<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="E21:20_1_1"/><i><font color="Green">E74</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="20_1_1_9"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_9_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/><b>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="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">EQR1</font> = <font color="Maroon">U2</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">U3</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">EQR2</font> = <font color="Maroon">U1</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">U2</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</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>,<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>,<font color="Maroon">EQR1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">U1</font> <a href="pboole.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="pboole.html#K3">/\</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="pboole.html#K3">/\</a> <font color="Maroon">U2</font></span>)</span> <a href="pboole.html#K3">/\</a> <font color="Maroon">U3</font>
<b>by </b><i><a class="ref" href="pboole.html#T35">PBOOLE:35</a></i>
<br/>.= 
<font color="Maroon">B2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">EQR2</font>,<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="E2:20_1_1_9"/>
<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="E22:20_1_1"/><i><font color="Green">E78</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="20_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>;<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="E23:20_1_1"/><i><font color="Green">E80</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="20_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>, <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="E2:20_1_1_11"/><i><font color="Green">E81</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="E24:20_1_1"/><i><font color="Green">E82</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="20_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/>
<a NAME="E2:20_1_1_12"/><i><font color="Green">E85</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="20_1_1_12_1"><b>proof </b></a><div class="add">

<b>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="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">EQR</font> = <font color="Maroon">U1</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">U2</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<a NAME="E3:20_1_1_12_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="pboole.html#K3">/\</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">EQR</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">U2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><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="E25:20_1_1"/><i><font color="Green">E86</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="20_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>;<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="E26:20_1_1"/><i><font color="Green">E87</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="20_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>, <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="E2:20_1_1_14"/><i><font color="Green">E88</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="msualg_5.html#T1" target="_self">Th1</a>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E27:20_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="20_1_1_15"><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="E2:20_1_1_15"/><i><font color="Green">E89</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="20_1_1_15_1"><b>proof </b></a><div class="add">

<b>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="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<a NAME="E2:20_1_1_15_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="msualg_5.html#K4" target="_self">"\/"</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="pboole.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">U2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><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="msualg_5.html#T1" target="_self">Th1</a>, <a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E28:20_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="E30:20_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> iff <font color="Olive">x</font> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_16"><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>thus </b><a NAME="E1:20_1_1_16"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> implies <font color="Maroon">x</font> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:20_1_1_16"/>
( <font color="Maroon">x</font> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_16_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:20_1_1_16_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">M2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:20_1_1_16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="msualg_5.html#T1" target="_self">Th1</a></i>;<br/>


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


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

<b>thus </b><a NAME="E31:20_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</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="pboole.html#K3">/\</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="msualg_5.html#K4" target="_self">"\/"</a> <font color="Olive">y</font> )
 <b>by </b><i>, </i>;<br/>


</div>
