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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:27_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></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> ] means <font color="Maroon">a<sub>1</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E5:27_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<i><font color="Green">E24</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:27_1_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></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">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="msualg_5.html#E2:27_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E4:27_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>7</sub></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">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>hence </b><a NAME="E5:27_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msualg_5.html#E3:27_1_1_1_2"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="msualg_5.html#E1:27_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:27_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:27_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_5.html#E4:27_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>

<b>thus </b><a NAME="E2:27_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:27_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_5.html#E5:27_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msualg_5.html#E4:27_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msualg_5.html#E1:27_1_1_2_1"><i><font color="Green">E25</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E8:27_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E10:27_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:27_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_5.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:27_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>11</sub></font> = the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E13:27_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:27_1_1_4"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_5.html#E1:27_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msualg_5.html#E3:27_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E6:27_1_1_4"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E7:27_1_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then </b><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msualg_5.html#E3:27_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>

<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5" target="_self">Def5</a></i>
;<br/><a NAME="E9:27_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msualg_5.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E10:27_1_1_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E13:27_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E16:27_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_5"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_5"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:27_1_1_5"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_5.html#E1:27_1_1_5"><i><font color="Green">E28</font></i></a>, <a class="txt" href="msualg_5.html#E3:27_1_1_5"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E6:27_1_1_5"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E7:27_1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then </b><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msualg_5.html#E3:27_1_1_5"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>

<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5" target="_self">Def5</a></i>
;<br/><a NAME="E9:27_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msualg_5.html#T18" target="_self">Th18</a></i>;<br/><b>hence </b><a NAME="E10:27_1_1_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E16:27_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a>  ;<br/>
<a NAME="E20:27_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#K3">/\</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:27_1_1_6"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E2:27_1_1_6"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msualg_5.html#E2:27_1_1_6"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>

<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5" target="_self">Def5</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msualg_5.html#E2:27_1_1_6"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>

<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5" target="_self">Def5</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E21:27_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="binop_1.html#V1">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1_7"/>
<font color="Maroon">c<sub>12</sub></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="E22:27_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="binop_1.html#V2">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>20</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#T17" target="_self">Th17</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#T17" target="_self">Th17</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> = 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="msualg_5.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1_8"/>
<font color="Maroon">c<sub>12</sub></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="E23:27_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="binop_1.html#V1">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_9"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1_9"/>
<font color="Maroon">c<sub>13</sub></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="E24:27_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="binop_1.html#V2">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_10"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#T18" target="_self">Th18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#T18" target="_self">Th18</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> = 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="pboole.html#T35">PBOOLE:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1_10"/>
<font color="Maroon">c<sub>13</sub></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="E25:27_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font>;<br/>

<b>thus </b><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E21:27_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>15</sub></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="E26:27_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font>;<br/>

<b>thus </b><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></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">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E22:27_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>17</sub></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="E27:27_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font>;<br/>

<a NAME="E1:27_1_1_13"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_13_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#T18" target="_self">Th18</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="ref" href="msualg_5.html#T12" target="_self">Th12</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E1:27_1_1_13"><i><font color="Green">E36</font></i></a>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E28:27_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font>;<br/>

<b>thus </b><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E23:27_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>15</sub></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="E29:27_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font>;<br/>

<b>thus </b><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> = 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></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">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E24:27_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>17</sub></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="E30:27_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font>;<br/>

<a NAME="E1:27_1_1_16"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_16_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#T17" target="_self">Th17</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> = 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_5.html#E20:27_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="msualg_5.html#T11" target="_self">Th11</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> = 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E1:27_1_1_16"><i><font color="Green">E38</font></i></a>, <a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E31:27_1_1"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="lattices.html#V5">join-associative</a> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="lattices.html#V8">meet-absorbing</a> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="lattices.html#V7">meet-associative</a> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i><a class="txt" href="msualg_5.html#E25:27_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msualg_5.html#E26:27_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msualg_5.html#E27:27_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msualg_5.html#E28:27_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msualg_5.html#E29:27_1_1"><i><font color="Green">E37</font></i></a>, <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">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <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>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as  <a href="lattices.html#V3">strict</a>  <a href="nat_lat.html#M2">SubLattice</a> of  <a href="msualg_5.html#K5" target="_self">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_5.html#E10:27_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_lat.html#D16">NAT_LAT:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E34:27_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_5.html#E6:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div>
