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

<span class="kw">set </span><font color="Maroon" title="c3">D</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">f</font> =  the   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p2">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">M2</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E2:25_1_1"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c5">M2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c6">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:25_1_1"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">M2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E5:25_1_1"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 for <font color="Olive" title="b1">C1</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Olive" title="b1">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">M2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">C1</font> be   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">M2</font></span><br/>

<div><span class="lab"><font color="Green" title="E23">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">M2</font> implies <font color="Maroon" title="c7">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">M2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">x9</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E3:25_1_1_1_1"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c7">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c9">x9</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x9</font></span>)</span>,<span class="p2">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x9</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><a NAME="E4:25_1_1_1_1"/>
<font color="Maroon" title="c5">M2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x9</font></span>)</span>,<span class="p2">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x9</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_1_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:25_1_1_1_1"/>
<font color="Maroon" title="c7">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_1_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">C1</font> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">M2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:25_1_1_1"/>
<font color="Maroon" title="c7">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:25_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> iff <font color="Olive" title="b1">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> iff <font color="Maroon" title="c7">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:25_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> implies <font color="Maroon" title="c7">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:25_1_1_2"/>
( <font color="Maroon" title="c7">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_1_1_2_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c7">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font></span><br/>

<a NAME="E2:25_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">M2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:25_1_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:25_1_1_2_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E7:25_1_1"/><span class="kw">then </span>
 the   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">B</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">B1</font> =  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">B2</font> =  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">B</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span></span><br/><a NAME="E2:25_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D5" target="_self" title="MSUALG_5:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E11:25_1_1"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">B1</font> =  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c11">B2</font> =  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">B</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>;<br/>
<a NAME="E13:25_1_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c11">B2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1_1_4"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">B2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:25_1_1_4"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E4:25_1_1_4"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_4"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E3:25_1_1_4"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E5:25_1_1_4"/>
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_4"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E3:25_1_1_4"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x19</font> = <font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c16">x29</font> = <font color="Maroon" title="c14">x2</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E4:25_1_1_4"><span class="lab"><font color="Green" title="E28">A12</font></span></a></span>;<br/><a NAME="E7:25_1_1_4"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c16">x29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><a NAME="E8:25_1_1_4"/>
<font color="Maroon" title="c15">x19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><a NAME="E9:25_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_1_1_4"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">B2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> = 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_4"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c15">x19</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c16">x29</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D5" target="_self" title="MSUALG_5:def.5">Def5</a></span>
;<br/><a NAME="E11:25_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">B2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T16" target="_self" title="MSUALG_5:th.16">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E12:25_1_1_4"/>
<font color="Maroon" title="c11">B2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">B2</font> = <font color="Maroon" title="c11">B2</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c7">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<a NAME="E16:25_1_1"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c10">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1_1_5"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c15">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:25_1_1_5"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E4:25_1_1_5"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_5"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E3:25_1_1_5"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E5:25_1_1_5"/>
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_5"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E3:25_1_1_5"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">x19</font> = <font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c17">x29</font> = <font color="Maroon" title="c15">x2</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E4:25_1_1_5"><span class="lab"><font color="Green" title="E29">A17</font></span></a></span>;<br/><a NAME="E7:25_1_1_5"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<font color="Maroon" title="c17">x29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><a NAME="E8:25_1_1_5"/>
<font color="Maroon" title="c16">x19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><a NAME="E9:25_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_1_1_5"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c10">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
 the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_5"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c16">x19</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c17">x29</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D5" target="_self" title="MSUALG_5:def.5">Def5</a></span>
;<br/><a NAME="E11:25_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T15" target="_self" title="MSUALG_5:th.15">Th17</a></span>;<br/><span class="kw">hence </span><a NAME="E12:25_1_1_5"/>
<font color="Maroon" title="c10">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">B1</font> = <font color="Maroon" title="c10">B1</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c7">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:25_1_1"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">L</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c13">B1</font>,<font color="Maroon" title="c12">B2</font> #) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a>  ;<br/>
<a NAME="E20:25_1_1"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Olive" title="b2">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> be   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c16">b</font> &amp; <font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c16">b</font> )</span><br/>

<a NAME="E1:25_1_1_6"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<a NAME="E2:25_1_1_6"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<a NAME="E3:25_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A21</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_6"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) = 
 the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c15">a</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c16">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D5" target="_self" title="MSUALG_5:def.5">Def5</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c16">b</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) = 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_6"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c15">a</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c16">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D5" target="_self" title="MSUALG_5:def.5">Def5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E28">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">a</font>)</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c17">a9</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c18">b9</font> = <font color="Maroon" title="c16">b</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) = 
<font color="Maroon" title="c17">a9</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c18">b9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E22:25_1_1"/><span class="lab"><font color="Green" title="E29">A23</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c15">a</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font> = 
<font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
 the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:25_1_1"><span class="lab"><font color="Green" title="E28">A22</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c15">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E30">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font>, <font color="Maroon" title="c17">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c17">c</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>,<font color="Maroon" title="c17">c</font>)</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c18">a9</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c19">b9</font> = <font color="Maroon" title="c16">b</font>, <font color="Maroon" title="c20">c9</font> = <font color="Maroon" title="c17">c</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">e1</font> = <font color="Maroon" title="c19">b9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c20">c9</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T16" target="_self" title="MSUALG_5:th.16">Th18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">e2</font> = <font color="Maroon" title="c18">a9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c19">b9</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T16" target="_self" title="MSUALG_5:th.16">Th18</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c17">c</font>)</span>)</span>) = 
<font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c21">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c18">a9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">b9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c20">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">a9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c19">b9</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c20">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T29" title="PBOOLE:th.29">PBOOLE:29</a></span>
<br/>.= 
<font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c22">e2</font>,<font color="Maroon" title="c17">c</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>,<font color="Maroon" title="c17">c</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E24:25_1_1"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font>, <font color="Maroon" title="c17">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">c</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">c</font></span>)</span> = 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">c</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c17">c</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"> the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>,<font color="Maroon" title="c17">c</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:25_1_1"><span class="lab"><font color="Green" title="E30">A24</font></span></a></span>
<br/>.= 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">b</font></span>)</span>,<font color="Maroon" title="c17">c</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E32">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_11"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font>, <font color="Maroon" title="c17">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c17">c</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>,<font color="Maroon" title="c17">c</font>)</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c18">a9</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c19">b9</font> = <font color="Maroon" title="c16">b</font>, <font color="Maroon" title="c20">c9</font> = <font color="Maroon" title="c17">c</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">d</font> = <font color="Maroon" title="c19">b9</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c20">c9</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T15" target="_self" title="MSUALG_5:th.15">Th17</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">e</font> = <font color="Maroon" title="c18">a9</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c19">b9</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T15" target="_self" title="MSUALG_5:th.15">Th17</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c17">c</font>)</span>)</span>) = 
<font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c21">d</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c18">a9</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">b9</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c20">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">a9</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c19">b9</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c20">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T8" target="_self" title="MSUALG_5:th.8">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c22">e</font>,<font color="Maroon" title="c17">c</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>,<font color="Maroon" title="c17">c</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E26:25_1_1"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font>, <font color="Maroon" title="c17">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">c</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">c</font></span>)</span> = 
 the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">c</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c17">c</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
 the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>,<font color="Maroon" title="c17">c</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:25_1_1"><span class="lab"><font color="Green" title="E32">A26</font></span></a></span>
<br/>.= 
 the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font></span>)</span>,<font color="Maroon" title="c17">c</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E34">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_13"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">a</font>)</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c17">a9</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c18">b9</font> = <font color="Maroon" title="c16">b</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>) = 
<font color="Maroon" title="c18">b9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c17">a9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E28:25_1_1"/><span class="lab"><font color="Green" title="E35">A29</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c15">a</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">b</font> = 
<font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:25_1_1"><span class="lab"><font color="Green" title="E34">A28</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c15">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:25_1_1"/><span class="lab"><font color="Green" title="E36">A30</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font></span><br/>

<a NAME="E1:25_1_1_15"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
<font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c17">a9</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c18">b9</font> = <font color="Maroon" title="c16">b</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">d</font> = <font color="Maroon" title="c17">a9</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c18">b9</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T15" target="_self" title="MSUALG_5:th.15">Th17</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>) = 
<font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c19">d</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">a9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a9</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c18">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c15">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T9" target="_self" title="MSUALG_5:th.9">Th11</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font></span>)</span> = 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c15">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c15">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_15"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:25_1_1"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_16"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font></span><br/>

<a NAME="E1:25_1_1_16"/><span class="lab"><font color="Green" title="E37">A32</font></span>: 
<font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>,<font color="Maroon" title="c16">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c17">a9</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c18">b9</font> = <font color="Maroon" title="c16">b</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">EQR</font> = <font color="Maroon" title="c17">a9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c18">b9</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T16" target="_self" title="MSUALG_5:th.16">Th18</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">B2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span>,<font color="Maroon" title="c16">b</font>) = 
<font color="Maroon" title="c13">B1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c17">a9</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c18">b9</font></span>)</span>,<font color="Maroon" title="c18">b9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c19">EQR</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c18">b9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T10" target="_self" title="MSUALG_5:th.10">Th12</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">b</font> = 
 the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c14">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">b</font></span>)</span>,<font color="Maroon" title="c16">b</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c16">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_16"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E31:25_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">L</font> is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>  &amp; <font color="Maroon" title="c14">L</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a>  &amp; <font color="Maroon" title="c14">L</font> is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a>  &amp; <font color="Maroon" title="c14">L</font> is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>  &amp; <font color="Maroon" title="c14">L</font> is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a>  &amp; <font color="Maroon" title="c14">L</font> is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:25_1_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a>, <a class="txt" href="#E26:25_1_1"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="txt" href="#E28:25_1_1"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E24:25_1_1"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="txt" href="#E29:25_1_1"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">L</font> = <font color="Maroon" title="c14">L</font> as   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">L</font> = <font color="Maroon" title="c15">L</font> as   <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a>   <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of  <a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25_1_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="nat_lat.html#D12" title="NAT_LAT:def.12">NAT_LAT:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">L</font> iff <font color="Olive" title="b1">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">thus </span><a NAME="E34:25_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">L</font> iff <font color="Olive" title="b1">x</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
