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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:68_1_1"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V3" title="COMPTS_1:attr.3">being_T2</a> )
 ;<br/>

<b>hence </b><a NAME="E2:68_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR3" title="COMPTS_1:NR.3">is_T3</a>  &amp; <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR4" title="COMPTS_1:NR.4">is_T4</a>  )
 <b>by </b><i><a class="ref" href="compts_1.html#T21" title="COMPTS_1:th.21">COMPTS_1:21</a>, <a class="ref" href="compts_1.html#T22" title="COMPTS_1:th.22">COMPTS_1:22</a></i>;<br/>

<a NAME="E3:68_1_1"/><i><font color="Green" title="E32">A3</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="waybel_3.html#E1:68_1_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="compts_1.html#T10" title="COMPTS_1:th.10">COMPTS_1:10</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9" target="_self" title="WAYBEL_3:def.9">WAYBEL_3:def 9</a><br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E4:68_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">Y</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:68_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:68_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="urysohn1.html#T29" title="URYSOHN1:th.29">URYSOHN1:29</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">Z</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">Y</font>;<br/>

<a NAME="E7:68_1_1"/>
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Z</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E8:68_1_1"/><b>then </b>
( <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">Z</font> &amp; <font color="Maroon" title="c5">Z</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T56" title="TOPS_1:th.56">TOPS_1:56</a></i>;<br/>
<b>hence </b><a NAME="E9:68_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">Z</font> &amp; <font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">Z</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:68_1_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="waybel_3.html#E6:68_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="compts_1.html#T18" title="COMPTS_1:th.18">COMPTS_1:18</a></i>;<br/>


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