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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>  implies <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V12" title="PRE_TOPC:attr.12">T_4</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_2_1"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V12" title="PRE_TOPC:attr.12">T_4</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_2_1_1"/>
( <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_2_1_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V12" title="PRE_TOPC:attr.12">T_4</a> </span><br/><div class="add"><a NAME="E2:9_2_1_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">F2</font> holds <br/> ex <font color="Olive" title="b3">G1</font>, <font color="Olive" title="b4">G2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">G1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b4">G2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b2">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">G2</font> &amp; <font color="Olive" title="b3">G1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">G2</font> )
 ;<br/><a NAME="E3:9_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V10" title="PRE_TOPC:attr.10">normal</a> 
 ;<br/><span class="kw">hence </span><a NAME="E4:9_2_1_1_1"/>
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V12" title="PRE_TOPC:attr.12">T_4</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2_1_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_2_1_1_2"/>
 not <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V12" title="PRE_TOPC:attr.12">T_4</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">t</font> = <font color="Maroon" title="c1">T</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/><a NAME="E3:9_2_1_1_2"/>
( <font color="Maroon" title="c2">t</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a>  &amp;  ex <font color="Olive" title="b1">Bn</font> being   <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c2">t</font> st <font color="Olive" title="b1">Bn</font> is  <a href="nagata_1.html#V6" title="NAGATA_1:attr.6">Basis_sigma_locally_finite</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_2.html#T19" onmouseover="rs('nagata_2/T19')" onmouseout="rh()">NAGATA_2:19</a></span>;<br/><a NAME="E4:9_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">t</font> is  <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V10" title="PRE_TOPC:attr.10">normal</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#T20" onmouseover="rs('nagata_1/T20')" onmouseout="rh()">NAGATA_1:20</a>, <a class="ref" href="nagata_2.html#T19" onmouseover="rs('nagata_2/T19')" onmouseout="rh()">NAGATA_2:19</a></span>;<br/><span class="kw">hence </span><a NAME="E5:9_2_1_1_2"/>
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V12" title="PRE_TOPC:attr.12">T_4</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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