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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a></span><br/>

<b>set </b><font color="Maroon" title="c2">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #);<br/>
<a NAME="E1:46"/><i><font color="Green" title="E30">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="rusub_5.html#T38" target="_self" title="RUSUB_5:th.38">Th38</a></i>;<br/>
<a NAME="E2:46"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #)  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="rusub_5.html#T40" target="_self" title="RUSUB_5:th.40">Th40</a></i>;<br/>
<a NAME="E3:46"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #)  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #) &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #) holds <br/><font color="Olive" title="b1">p</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="rusub_5.html#T39" target="_self" title="RUSUB_5:th.39">Th39</a></i>;<br/>
<b>hence </b><a NAME="E4:46"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>
 <b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E2:46"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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