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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">Vt</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Vt</font>
 ;<br/>

<a NAME="E2:31"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c3">Vt</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="normsp_2.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></i>;<br/>
<a NAME="E3:31"/>
( <font color="Maroon" title="c3">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Maroon" title="c3">Vt</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>
<a NAME="E4:31"/><b>then </b>
( <font color="Maroon" title="c3">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Maroon" title="c2">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="normsp_2.html#E2:31"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="normsp_2.html#T20" target="_self" title="NORMSP_2:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E5:31"/>
( <font color="Maroon" title="c2">V</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Maroon" title="c3">Vt</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#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>


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