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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/>( <font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  iff <font color="Olive" title="b1">B</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">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> implies ( <font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  iff <font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:82"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  iff <font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:82"/>
( <font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  implies <font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  implies <font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:82_1"/>
<font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<a NAME="E2:82_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 
;<br/>
<a NAME="E3:82_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D5" title="RCOMP_1:def.5">RCOMP_1:def 5</a></span>;<br/>
<a NAME="E4:82_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5a.html#T5" title="JORDAN5A:th.5">JORDAN5A:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:82_1"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B</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="#E1:82"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="topreal6.html#E71"><span class="lab"><font color="Green" title="E36">Lm2</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a>, <a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pre_topc.html#D3" title="PRE_TOPC:def.3">PRE_TOPC:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:82"/>
<font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> </span><br/>

<a NAME="E4:82"/><span class="kw">then </span>
<font color="Maroon" title="c2">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E5:82"/><span class="kw">then </span>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="topreal6.html#E71"><span class="lab"><font color="Green" title="E36">Lm2</font></span></a>, <a class="ref" href="jordan5a.html#T5" title="JORDAN5A:th.5">JORDAN5A:5</a>, <a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/>
<a NAME="E6:82"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D5" title="RCOMP_1:def.5">RCOMP_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:82"/>
<font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 ; <a class="txt" 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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