<?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">S</font> being  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">S9</font> being  <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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Olive" title="b2">S9</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">S</font> holds <br/>( ( <font color="Olive" title="b1">S</font> is  <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a>  implies <font color="Olive" title="b2">S9</font> is  <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a>  ) &amp; ( <font color="Olive" title="b2">S9</font> is  <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a>  implies <font color="Olive" title="b1">S</font> is  <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a>  ) &amp; ( <font color="Olive" title="b1">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  implies <font color="Olive" title="b2">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  ) &amp; ( <font color="Olive" title="b2">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  implies <font color="Olive" title="b1">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  ) )</span><br/><span class="kw">let </span><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>; <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">S9</font> being  <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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Olive" title="b1">S9</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">S</font> holds <br/>( ( <font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a>  implies <font color="Olive" title="b1">S9</font> is  <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a>  ) &amp; ( <font color="Olive" title="b1">S9</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>  ) &amp; ( <font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  implies <font color="Olive" title="b1">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  ) &amp; ( <font color="Olive" title="b1">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S9</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#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">S9</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">S</font> implies ( ( <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">S9</font> is  <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a>  ) &amp; ( <font color="Maroon" title="c3">S9</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>  ) &amp; ( <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">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  ) &amp; ( <font color="Maroon" title="c3">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c3">S9</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">S</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">S</font> is  <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a>  implies <font color="Maroon" title="c3">S9</font> is  <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a>  ) &amp; ( <font color="Maroon" title="c3">S9</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>  ) &amp; ( <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">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  ) &amp; ( <font color="Maroon" title="c3">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  ) )</span><br/>

<span class="kw">thus </span><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">S9</font> is  <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">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">S9</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>  ) &amp; ( <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">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  ) &amp; ( <font color="Maroon" title="c3">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">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">S9</font> is  <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a> </span><br/>

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="compl_sp.html#D5" target="_self" title="COMPL_SP:def.5">COMPL_SP:def 5</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">S9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:20_1"/>
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</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="#E1:20_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="compl_sp.html#D7" target="_self" title="COMPL_SP:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_1"/>
<font color="Maroon" title="c3">S9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</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="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</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="E3:20"/>
( <font color="Maroon" title="c3">S9</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>  )
  <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">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  iff <font color="Maroon" title="c3">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_2"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c3">S9</font> is  <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">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">S</font> is  <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a> </span><br/>

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="compl_sp.html#D7" target="_self" title="COMPL_SP:def.7">COMPL_SP:def 7</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">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</font> is  <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a> </span><br/>

<a NAME="E2:20_2"/>
<font color="Maroon" title="c3">S9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</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="#E1:20_2"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="compl_sp.html#D5" target="_self" title="COMPL_SP:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_2"/>
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</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="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</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="E4:20"/>
( <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">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">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">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_3"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">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">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> </span><br/>

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="compl_sp.html#D6" target="_self" title="COMPL_SP:def.6">COMPL_SP:def 6</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">S9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<a NAME="E2:20_3"/>
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</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="txt" href="#E1:20_3"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="compl_sp.html#D8" target="_self" title="COMPL_SP:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_3"/>
<font color="Maroon" title="c3">S9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</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="txt" href="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</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="E5:20"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c3">S9</font> is  <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">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">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> </span><br/>

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="compl_sp.html#D8" target="_self" title="COMPL_SP:def.8">COMPL_SP:def 8</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">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</font> is  <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a> </span><br/>

<a NAME="E6:20"/>
<font color="Maroon" title="c3">S9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</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="txt" href="#E5:20"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="compl_sp.html#D6" target="_self" title="COMPL_SP:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:20"/>
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">i</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="txt" href="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</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>
