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<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">R</font>  st the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">T</font> <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">R</font> holds <br/><font color="Olive" title="b1">T</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a></span><br/><b>let </b><font color="Maroon" title="c2">T</font> be    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> <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">R</font> implies <font color="Maroon" title="c2">T</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> )</span><br/>



<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E35">A1</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> <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">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">T</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a></span><br/>

<a NAME="E2:75"/><i><font color="Green" title="E36">A2</font></i>: 
 <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> #) <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="c1">R</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" target="_self" title="YELLOW_9:def.4">Def4</a></i>;<br/>
<a NAME="E3:75"/>
<font color="Maroon" title="c2">T</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  not <font color="Maroon" title="c3">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or ( <font color="Maroon" title="c3">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> ) ) &amp; (  not <font color="Maroon" title="c3">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> or  not <font color="Maroon" title="c3">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> or <font color="Maroon" title="c3">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="#E2:75"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E2:75_1"/>
( <font color="Maroon" title="c3">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies ( <font color="Maroon" title="c3">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> or  not <font color="Maroon" title="c3">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> or <font color="Maroon" title="c3">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1"/>
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )</span><br/>

<a NAME="E2:75_1_1"/><b>then </b>
( <font color="Maroon" title="c4">A</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 
<b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="waybel11.html#T31" title="WAYBEL11:th.31">WAYBEL11:31</a></i>;<br/>
<b>hence </b><a NAME="E3:75_1_1"/>
( <font color="Maroon" title="c3">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 <b>by </b><i><a class="txt" href="#E2:75"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="yellow_9.html#T47" target="_self" title="YELLOW_9:th.47">Th47</a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:75_1"/>
( <font color="Maroon" title="c3">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<a NAME="E4:75_1"/><b>then </b>
( <font color="Maroon" title="c4">A</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 
<b>by </b><i><a class="txt" href="#E2:75"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="yellow_9.html#T47" target="_self" title="YELLOW_9:th.47">Th47</a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E5:75_1"/>
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="waybel11.html#T31" title="WAYBEL11:th.31">WAYBEL11:31</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:75"/>
<font color="Maroon" title="c2">T</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>
 ; <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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