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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#V3" title="RCOMP_1:attr.3">open</a> iff <font color="Olive" title="b1">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#V3" title="RCOMP_1:attr.3">open</a> iff <font color="Maroon" title="c2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )</span><br/>



<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E41">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> iff <font color="Maroon" title="c2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/><a NAME="E2:64_1"/><b>then </b>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> 
 <b>by </b><i><a class="ref" href="jordan5a.html#T6" title="JORDAN5A:th.6">JORDAN5A:6</a></i>;<br/><a NAME="E3:64_1"/><b>then </b>
<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/><b>hence </b><a NAME="E4:64_1"/>
<font color="Maroon" title="c2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:64"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:64"/>
<font color="Maroon" title="c2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a></span><br/>

<a NAME="E4:64"/><b>then </b>
<font color="Maroon" title="c2">B</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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:64"/><b>then </b>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> 
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:64"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:64"/>
<font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 <b>by </b><i><a class="ref" href="jordan5a.html#T6" title="JORDAN5A:th.6">JORDAN5A:6</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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