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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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</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_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>;<br/>



<a NAME="E1:20"/><i><font color="Green">E36</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E2:20"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:20"/>
( <font color="Maroon">x'</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">y'</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:20"/><b>then </b>
<font color="Maroon">x'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">y'</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/>
<a NAME="E6:20"/><b>then </b>
<font color="Maroon">x'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">y'</font> <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#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:20"/>
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T8">YELLOW_1:8</a></i>;<br/>

<a NAME="E8:20"/>
<font color="Maroon">x'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">y'</font> <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#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:20"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>


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