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<b>let </b><font color="Maroon" title="c1">M</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">A'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</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">A</font> holds <br/>( ( <font color="Olive" title="b1">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> implies <font color="Olive" title="b2">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) &amp; ( <font color="Olive" title="b2">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Olive" title="b1">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> ) &amp; ( <font color="Olive" title="b1">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> implies <font color="Olive" title="b2">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) &amp; ( <font color="Olive" title="b2">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Olive" title="b1">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>; <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">A'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</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">A</font> holds <br/>( ( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> implies <font color="Olive" title="b1">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) &amp; ( <font color="Olive" title="b1">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> ) &amp; ( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> implies <font color="Olive" title="b1">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) &amp; ( <font color="Olive" title="b1">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">A'</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">A</font> implies ( ( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> implies <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) &amp; ( <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> ) &amp; ( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> implies <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) &amp; ( <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> ) ) )</span><br/>





<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> implies <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) &amp; ( <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> ) &amp; ( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> implies <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) &amp; ( <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> ) )</span><br/>

<b>thus </b><a NAME="E2:13"/>
( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> implies <font color="Maroon" title="c3">A'</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="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> ) &amp; ( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> implies <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) &amp; ( <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">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="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<a NAME="E2:13_1"/><b>then </b>
<font color="Maroon" title="c2">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> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="compl_sp.html#D3" target="_self" title="COMPL_SP:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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>thus </b><a NAME="E3:13"/>
( <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">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">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> iff <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_2"/>
<font color="Maroon" title="c3">A'</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="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a></span><br/>

<a NAME="E2:13_2"/><b>then </b>
<font color="Maroon" title="c3">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> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2"/>
<font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="compl_sp.html#D3" target="_self" title="COMPL_SP:def.3">Def3</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>thus </b><a NAME="E4:13"/>
( <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> implies <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3"/>
<font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<a NAME="E2:13_3"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a>
 
<b>by </b><i><a class="ref" href="compl_sp.html#D4" target="_self" title="COMPL_SP:def.4">Def4</a></i>;<br/>
<a NAME="E3:13_3"/><b>then </b>
<font color="Maroon" title="c2">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#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="compl_sp.html#D3" target="_self" title="COMPL_SP:def.3">Def3</a></i>;<br/>
<a NAME="E4:13_3"/><b>then </b>
<font color="Maroon" title="c3">A'</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="compl_sp.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:13_3"/>
<font color="Maroon" title="c3">A'</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>; <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="E5:13"/>
<font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a></span><br/>

<a NAME="E6:13"/><b>then </b>
<font color="Maroon" title="c3">A'</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="E7:13"/><b>then </b>
<font color="Maroon" title="c2">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#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:13"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a>
 
<b>by </b><i><a class="ref" href="compl_sp.html#D3" target="_self" title="COMPL_SP:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E9:13"/>
<font color="Maroon" title="c2">A</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a>
 <b>by </b><i><a class="ref" href="compl_sp.html#D4" target="_self" title="COMPL_SP:def.4">Def4</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>
