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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel33.html#V1" title="WAYBEL33:attr.1">lim-inf</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">F</font> be  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c3">x</font> =  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c2">F</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>take </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font>;<br/><b>thus </b><a NAME="E2:41_1"/>
<font color="Maroon" title="c4">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="waybel33.html#T26" target="_self" title="WAYBEL33:th.26">Th26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41"/>
<font color="Maroon" title="c1">L</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="ref" href="yellow19.html#T33" title="YELLOW19:th.33">YELLOW19:33</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c2">T</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a> <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">L</font>;<br/><b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E2:41_2"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> #)
 <b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> ;<br/><a NAME="E4:41_2"/>
<font color="Maroon" title="c1">L</font> is    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#T44" title="YELLOW_9:th.44">YELLOW_9:44</a></i>;<br/><a NAME="E5:41_2"/><b>then </b>
( <font color="Maroon" title="c1">L</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="ref" href="waybel33.html#T25" target="_self" title="WAYBEL33:th.25">Th25</a></i>;<br/><a NAME="E6:41_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="waybel33.html#T23" target="_self" title="WAYBEL33:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E7:41_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:41"/>
<font color="Maroon" title="c1">L</font> is <a href="urysohn1.html#V1" title="URYSOHN1:attr.1">being_T1</a>
 <b>by </b><i><a class="ref" href="frechet2.html#T47" title="FRECHET2:th.47">FRECHET2:47</a></i>;<br/>


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