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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="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_2.html#K5" title="WAYBEL_2:func.5">sup_op</a> <font color="Maroon" title="c1">L</font> is <a href="waybel14.html#V1" title="WAYBEL14:attr.1">jointly_Scott-continuous</a> implies <font color="Maroon" title="c1">L</font> is <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">sober</a> )</span><br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="waybel_2.html#K5" title="WAYBEL_2:func.5">sup_op</a> <font color="Maroon" title="c1">L</font> is <a href="waybel14.html#V1" title="WAYBEL14:attr.1">jointly_Scott-continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">sober</a></span><br/>

<a NAME="E2:37"/><i><font color="Green" title="E28">A2</font></i>: 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#D12" title="WAYBEL11:def.12">WAYBEL11:def 12</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green" title="E29">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:37"/><i><font color="Green" title="E30">A4</font></i>: 
 <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">L</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T32" title="WAYBEL11:th.32">WAYBEL11:32</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">S</font> be  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D6" title="YELLOW_8:def.6">YELLOW_8:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<a NAME="E5:37"/><i><font color="Green" title="E31">A5</font></i>: 
( not <font color="Maroon" title="c2">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="ref" href="yellow_8.html#D4" title="YELLOW_8:def.4">YELLOW_8:def 4</a></i>;<br/>
<a NAME="E6:37"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>
<a NAME="E7:37"/><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel14.html#T2" target="_self" title="WAYBEL14:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">V</font> = <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E2:37"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="#E3:37"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="#E4:37"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="#E6:37"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E10:37"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E7:37"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="waybel14.html#T29" target="_self" title="WAYBEL14:th.29">Th29</a>, <a class="txt" href="#E2:37"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="#E4:37"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="waybel14.html#T17" target="_self" title="WAYBEL14:th.17">Th17</a></i>;<br/>
<a NAME="E11:37"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="#E10:37"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="tops_1.html#T21" title="TOPS_1:th.21">TOPS_1:21</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<a NAME="E12:37"/><i><font color="Green" title="E36">A10</font></i>: 
 <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="c4">p</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>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T9" title="WAYBEL11:th.9">WAYBEL11:9</a></i>;<br/>
<b>hence </b><a NAME="E13:37"/>
<font color="Maroon" title="c4">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="yellow_8.html#T27" title="YELLOW_8:th.27">YELLOW_8:27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c5">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">q</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> )</span><br/>

<b>assume </b><a NAME="E14:37"/>
<font color="Maroon" title="c5">q</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font></span><br/>

<a NAME="E15:37"/><b>then </b><i><font color="Green" title="E37">A11</font></i>: 
 <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="c5">q</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> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E5:37"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="yellow_8.html#T25" title="YELLOW_8:th.25">YELLOW_8:25</a></i>;<br/>
<a NAME="E16:37"/>
<font color="Maroon" title="c1">L</font> is <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">T_0</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T10" title="WAYBEL11:th.10">WAYBEL11:10</a></i>;<br/>
<b>hence </b><a NAME="E17:37"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="#E12:37"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="#E15:37"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="yellow_8.html#T32" title="YELLOW_8:th.32">YELLOW_8:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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