<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Olive" title="b2">A9</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">A9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) &amp; ( <font color="Olive" title="b2">A9</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">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) &amp; ( <font color="Olive" title="b2">A9</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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A9</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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Olive" title="b1">A9</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">A9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) &amp; ( <font color="Olive" title="b1">A9</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">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) &amp; ( <font color="Olive" title="b1">A9</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/><span class="kw">let </span><font color="Maroon" title="c3">A9</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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</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">A9</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">A9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) &amp; ( <font color="Maroon" title="c3">A9</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">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) &amp; ( <font color="Maroon" title="c3">A9</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/>





<span class="kw">assume </span><a NAME="E1:13"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c3">A9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) &amp; ( <font color="Maroon" title="c3">A9</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">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) &amp; ( <font color="Maroon" title="c3">A9</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/>

<span class="kw">thus </span><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">A9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
  <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">A9</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">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) &amp; ( <font color="Maroon" title="c3">A9</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:13_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <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> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D3" target="_self" title="COMPL_SP:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:13_1"/>
<font color="Maroon" title="c3">A9</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="#E1:13"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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/>


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

<span class="kw">thus </span><a NAME="E3:13"/>
( <font color="Maroon" title="c3">A9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:13_2"/>
<font color="Maroon" title="c3">A9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <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="c2">A</font> is  <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> </span><br/>

<a NAME="E2:13_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">A9</font> <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> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="compl_sp.html#D3" target="_self" title="COMPL_SP:def.3">Def3</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/>


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

<span class="kw">thus </span><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">A9</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A9</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<a NAME="E2:13_3"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D4" target="_self" title="COMPL_SP:def.4">Def4</a></span>;<br/>
<a NAME="E3:13_3"/><span class="kw">then </span>
<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#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D3" target="_self" title="COMPL_SP:def.3">Def3</a></span>;<br/>
<a NAME="E4:13_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">A9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:13_3"/>
<font color="Maroon" title="c3">A9</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="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</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/>


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

<span class="kw">assume </span><a NAME="E5:13"/>
<font color="Maroon" title="c3">A9</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="Red">::  thesis: </font></span></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"/><span class="kw">then </span>
<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#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E7:13"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D3" target="_self" title="COMPL_SP:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:13"/>
<font color="Maroon" title="c2">A</font> is  <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D4" target="_self" title="COMPL_SP:def.4">Def4</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/>


</div>
