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

<b>let </b><font color="Maroon" title="c1">M</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c3">S'</font> be   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</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>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">S'</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>thus </b><a NAME="E2:20"/>
( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a> implies <font color="Maroon" title="c3">S'</font> is <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compl_sp.html#D5" target="_self" title="COMPL_SP:def.5">COMPL_SP:def 5</a><br/>

<a NAME="E2:20_1"/>
<font color="Maroon" title="c2">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</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#E1:20_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="compl_sp.html#D7" target="_self" title="COMPL_SP:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1"/>
<font color="Maroon" title="c3">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</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#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:20"/>
( <font color="Maroon" title="c3">S'</font> is <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a> implies <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c3">S'</font> is <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compl_sp.html#D7" target="_self" title="COMPL_SP:def.7">COMPL_SP:def 7</a><br/>

<a NAME="E2:20_2"/>
<font color="Maroon" title="c3">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</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#E1:20_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="compl_sp.html#D5" target="_self" title="COMPL_SP:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:20_2"/>
<font color="Maroon" title="c2">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</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#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:20"/>
( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> implies <font color="Maroon" title="c3">S'</font> is <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_3"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compl_sp.html#D6" target="_self" title="COMPL_SP:def.6">COMPL_SP:def 6</a><br/>

<a NAME="E2:20_3"/>
<font color="Maroon" title="c2">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:20_3"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="compl_sp.html#D8" target="_self" title="COMPL_SP:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:20_3"/>
<font color="Maroon" title="c3">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:20"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c3">S'</font> is <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compl_sp.html#D8" target="_self" title="COMPL_SP:def.8">COMPL_SP:def 8</a><br/>

<a NAME="E6:20"/>
<font color="Maroon" title="c3">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E5:20"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="compl_sp.html#D6" target="_self" title="COMPL_SP:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E7:20"/>
<font color="Maroon" title="c2">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</a></i>;<br/>


</div>
