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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>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>;<br/>





<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/>
<a NAME="E1:90"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/>
<a NAME="E2:90"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E4:90"/>
<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#K7">~</a>  <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#K6">~</a> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D5">LATTICE3:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<a NAME="E6:90"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K8">~</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattice3.html#K8">~</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>;<br/>
<a NAME="E7:90"/>
( <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> iff <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E8:90"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> iff <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>


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