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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>assume </b><a NAME="E1:68_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">T</font> is <a href="compts_1.html#V2">compact</a> &amp; <font color="Maroon">T</font> is <a href="compts_1.html#V3">being_T2</a> )
 ;<br/>

<b>hence </b><a NAME="E2:68_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">T</font> <a href="compts_1.html#NR3" target="_self">is_T3</a>  &amp; <font color="Maroon">T</font> <a href="compts_1.html#NR4" target="_self">is_T4</a>  )
 <b>by </b><i><a class="ref" href="compts_1.html#T21">COMPTS_1:21</a>, <a class="ref" href="compts_1.html#T22">COMPTS_1:22</a></i>;<br/>

<a NAME="E3:68_1_1"/><i><font color="Green">E39</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="compts_1.html#V6">compact</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="compts_1.html#T10">COMPTS_1:10</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9" target="_self">WAYBEL_3:def 9</a><br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E4:68_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:68_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <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="urysohn1.html#T29">URYSOHN1:29</a></i>;<br/>
<b>take </b><font color="Maroon">Z</font> =  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Y</font>;<br/>

<a NAME="E7:68_1_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E8:68_1_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<b>hence </b><a NAME="E9:68_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z</font> is <a href="compts_1.html#V6">compact</a> )
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel_3.html#T1" target="_self">Th1</a>, <a class="ref" href="compts_1.html#T18">COMPTS_1:18</a></i>;<br/>


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