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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">Vt</font> being   <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>  st <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Vt</font> holds <br/>( <font color="Olive" title="b1">V</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  iff <font color="Olive" title="b2">Vt</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Vt</font> being   <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>  st <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Vt</font> holds <br/>( <font color="Maroon" title="c2">V</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  iff <font color="Olive" title="b1">Vt</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/><span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Vt</font> implies ( <font color="Maroon" title="c2">V</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  iff <font color="Maroon" title="c3">Vt</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">VVt</font> = <font color="Maroon" title="c3">Vt</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:44"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Vt</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  iff <font color="Maroon" title="c3">Vt</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E3:44"/>
( <font color="Maroon" title="c3">Vt</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c4">VVt</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#T20" target="_self" onmouseover="rs('normsp_2/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:44"/>
( <font color="Maroon" title="c2">V</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  iff <font color="Maroon" title="c3">Vt</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="normsp_2.html#T16" target="_self" onmouseover="rs('normsp_2/T16')" onmouseout="rh()">Th16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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