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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b2">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V</font> holds <br/>( <font color="Olive" title="b2">A</font> is  <a href="seq_4.html#V2" title="SEQ_4:attr.2">closed</a>  iff <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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 <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> holds <br/>( <font color="Olive" title="b1">A</font> is  <a href="seq_4.html#V2" title="SEQ_4:attr.2">closed</a>  iff <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> implies ( <font color="Maroon" title="c3">A</font> is  <a href="seq_4.html#V2" title="SEQ_4:attr.2">closed</a>  iff <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:6"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is  <a href="seq_4.html#V2" title="SEQ_4:attr.2">closed</a>  iff <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>

<a NAME="E2:6"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="seq_4.html#V1" title="SEQ_4:attr.1">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp1.html#T6" target="_self" title="COMPLSP1:th.6">Th108</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:6"/>
( <font color="Maroon" title="c3">A</font> is  <a href="seq_4.html#V2" title="SEQ_4:attr.2">closed</a>  iff <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T149" title="SEQ_4:th.149">SEQ_4:149</a>, <a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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