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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/>
<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>, <font color="Maroon">q</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">FILTER_0:def 7</a><br/>

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font> ;<br/>
<a NAME="E2:12"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">q'</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r'</font> =  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">q'</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r'</font> 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/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<a NAME="E5:12"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">p'</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:12"/>
<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">p'</font> <a href="openlatt.html#K3" target="_self">/\</a> <font color="Maroon">r'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">r1</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>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">r1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">r1'</font> = <font color="Maroon">r2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E10:12"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">r1'</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E11:12"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E12:12"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">r1'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:12"/><b>then </b>
<font color="Maroon">p'</font> <a href="openlatt.html#K3" target="_self">/\</a> <font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:12"/><b>then </b>
<font color="Maroon">r1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">r'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E15:12"/>
<font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


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