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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">A</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/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:66"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:66"/><i><font color="Green">E26</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/>
<b>assume </b><b>that </b><br/><a NAME="E3:66"/>
( not <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E4:66"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">S1</font>, <font color="Olive">S2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">S1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">S2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Olive">S1</font> <a href="finsub_1.html#K1">\/</a> <font color="Olive">S2</font> &amp; not <font color="Olive">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  holds <br/><font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D4">YELLOW_8:def 4</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D2">WAYBEL_6:def 2</a><br/>

<b>assume </b><a NAME="E5:66"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E6:66"/>
( <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> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E8:66"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">y1</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="E9:66"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x1</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">y1</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E10:66"/>
<font color="Maroon">x1</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">y1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E11:66"/><b>then </b>
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</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/>
<a NAME="E12:66"/><b>then </b>
<font color="Maroon">A</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="waybel12.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>
<a NAME="E13:66"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel12.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E14:66"/><b>then </b>
( <font color="Maroon">x1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  or <font color="Maroon">y1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="waybel12.html#T3" target="_self">Th3</a>, <a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:66"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="waybel12.html#T2" target="_self">Th2</a>, <a class="ref" href="tops_1.html#T21">TOPS_1:21</a></i>;<br/>


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