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<b>let </b><font color="Maroon" title="c1">S</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> #);<br/>
<a NAME="E1:81"/><i><font color="Green" title="E38">A1</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1"/><i><font color="Green" title="E39">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:81_1"/><b>then </b>
( the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T1" title="ZFMISC_1:th.1">ZFMISC_1:1</a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><a NAME="E3:81_1"/><b>then </b>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:81"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="yellow_9.html#E1:81_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="yellow_9.html#T11" target="_self" title="YELLOW_9:th.11">Th11</a>, <a class="ref" href="yellow_9.html#T17" target="_self" title="YELLOW_9:th.17">Th17</a></i>;<br/><a NAME="E4:81_1"/><b>then </b>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:81"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="yellow_9.html#T11" target="_self" title="YELLOW_9:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E5:81_1"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> #) is  <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> 
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:81_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="tdlat_3.html#D1" title="TDLAT_3:def.1">TDLAT_3:def 1</a>, <a class="ref" href="zfmisc_1.html#T1" title="ZFMISC_1:th.1">ZFMISC_1:1</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> #) as  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <b>by </b><i><a class="ref" href="cantor_1.html#T17" title="CANTOR_1:th.17">CANTOR_1:17</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">T</font>
;<br/>

<a NAME="E4:81"/>
( the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#T1" title="CANTOR_1:th.1">CANTOR_1:1</a>, <a class="ref" href="cantor_1.html#T4" title="CANTOR_1:th.4">CANTOR_1:4</a></i>;<br/>
<a NAME="E5:81"/><b>then </b><i><font color="Green" title="E39">A3</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:81"/>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T22" target="_self" title="YELLOW_9:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E7:81"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c3">T</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E5:81"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="cantor_1.html#D5" title="CANTOR_1:def.5">CANTOR_1:def 5</a></i>;<br/>


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