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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E2:47_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E3:47_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="tops_1.html#E1:47_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:47_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="tops_1.html#E2:47_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_1.html#T39" target="_self">Th39</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<a NAME="E3:47"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<a NAME="E4:47"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E5:47"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E2:47"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E6:47"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T39" target="_self">Th39</a></i>;<br/>
<b>hence </b><a NAME="E7:47"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tops_1.html#E4:47"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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