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<div class="add">

<a NAME="E1:4_1_1"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> st <br/>for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> holds <br/> ( <font color="Olive">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c'</font> =  <a href="msualg_3.html#K2">id</a> <font color="Maroon">M</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">a'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="eqrel_1.html#K2">/\</a> <font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="eqrel_1.html#T17">EQREL_1:17</a></i>
<br/>.= 
<font color="Maroon">c'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> = 
the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">c'</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>
;<br/>

<b>hence </b><a NAME="E6:4_1_1_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:4_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font> is <a href="lattices.html#V13">lower-bounded</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a></i>;<br/>
<a NAME="E3:4_1_1"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> st <br/>for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> holds <br/> ( <font color="Olive">c</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c'</font> = <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_1_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>consider </b><font color="Maroon">K1</font> being   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:4_1_1_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">K1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c'</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">c'</font> <a href="msualg_5.html#K4">"\/"</a> <font color="Maroon">a'</font> <a href="pboole.html#R6">=</a>  <a href="msualg_5.html#K3">EqCl</a> <font color="Maroon">K1</font> )
 <b>by </b><i><a class="ref" href="msualg_5.html#D4">MSUALG_5:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">K2</font> = <font color="Maroon">c'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font>, <font color="Maroon">a2</font> = <font color="Maroon">a'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">a'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D7">PBOOLE:def 7</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1">nabla</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="eqrel_1.html#D1">EQREL_1:def 1</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1">nabla</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_5.html#T4">MSUALG_5:4</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="eqrel_1.html#D1">EQREL_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="msualg_5.html#K4">"\/"</a> <font color="Maroon">a'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="msualg_5.html#K1">EqCl</a> <font color="Maroon">K2</font>
<b>by </b><i>, <a class="ref" href="msualg_5.html#D3">MSUALG_5:def 3</a></i>
<br/>.= 
<font color="Maroon">c'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="msualg_5.html#T3">MSUALG_5:3</a></i>
;<br/><br/></div><b>end;</b></div>
<b>thus </b><font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> = 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">c'</font> <a href="msualg_5.html#K4">"\/"</a> <font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>
;<br/>

<b>hence </b><a NAME="E6:4_1_1_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:4_1_1"/><b>then </b>
 <a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font> is <a href="lattices.html#V14">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:4_1_1"/>
 <a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font> is <a href="lattices.html#V15">bounded</a>
 <b>by </b><i>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>


</div>
