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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/>

<b>set </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #);<br/>
<a NAME="E1:46"/><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #) <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i/>;<br/>
<a NAME="E2:46"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #)  st <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #) holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="rusub_5.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:46"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #)  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #) &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #) holds <br/><font color="Olive">p</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:46"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font></span>)</span> #) is   <a href="pre_topc.html#NM1">TopSpace</a>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>


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