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<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:25_1"/><b>then </b>
<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">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:25_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">C</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">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><b>take </b><font color="Maroon">C</font> = <font color="Maroon">C</font>;<br/><b>thus </b><a NAME="E6:25_1"/>
( <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E2:25"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E3:25"/>
( <font color="Maroon">C</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">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:25"/><b>then </b>
<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">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:25"/>
<font color="Maroon">B</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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