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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">OL</font> =  <a href="openlatt.html#K6" target="_self">Open_setLatt</a> <font color="Maroon">T</font>;<br/>
<a NAME="E1:13"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a></i>;<br/>
<a NAME="E2:13"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">r</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K6" target="_self">Open_setLatt</a> <font color="Maroon">T</font></span>)</span> ;<br/>
<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K6" target="_self">Open_setLatt</a> <font color="Maroon">T</font></span>)</span>;<br/><b>thus </b><font color="Maroon">r</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p</font> = 
<font color="Maroon">r</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font>

;<br/><b>hence </b><a NAME="E2:13_1"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:13"/>
 <a href="openlatt.html#K6" target="_self">Open_setLatt</a> <font color="Maroon">T</font> is <a href="lattices.html#V13">lower-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">r</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a><br/>

<b>thus </b><a NAME="E1:13_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K6" target="_self">Open_setLatt</a> <font color="Maroon">T</font></span>)</span> holds <br/> ( <font color="Maroon">r</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>


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

<b>hence </b><a NAME="E6:13"/>
 <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="openlatt.html#K6" target="_self">Open_setLatt</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="lattices.html#D16">LATTICES:def 16</a></i>;<br/>


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