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<b>let </b><font color="Maroon">X</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">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">D</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/>

<b>assume </b><a NAME="E2:1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>

<a NAME="E3:1"/>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="tmap_1.html#K5">-extension_of_the_topology_of</a> <font color="Maroon">X</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="tmap_1.html#K5">-extension_of_the_topology_of</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T99">TMAP_1:99</a>, <a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<b>hence </b><a NAME="E4:1"/>
<font color="Maroon">C</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/>


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