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<b>thus </b><a NAME="E1:11_1_1"/>
( <font color="Maroon">L</font> is <a href="lattice3.html#V4">complete</a> implies for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">a</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="lattices.html#R1">[=</a> <font color="Olive">b</font> ) ) )
 <b>by </b><i><a class="ref" href="lattice3.html#D18">LATTICE3:def 18</a></i>;<br/>

<b>assume </b><a NAME="E2:11_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">a</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="lattices.html#R1">[=</a> <font color="Olive">b</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D18">LATTICE3:def 18</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span> ;<br/>
<a NAME="E3:11_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:11_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E6:11_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">r</font> holds <br/><font color="Maroon">p</font> <a href="lattices.html#R1">[=</a> <font color="Olive">r</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E7:11_1_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:11_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:11_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:11_1_1_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>


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

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

<b>assume </b><a NAME="E8:11_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:11_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:11_1_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/><b>thus </b><a NAME="E4:11_1_1_2"/>
<font color="Maroon">q</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:11_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E11:11_1_1"/>
<font color="Maroon">p</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


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