<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>   <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a>  <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</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">  for <font color="Olive" title="b1">T</font> being   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font> holds   <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</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 <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font>; <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">  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</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 <font color="Maroon" title="c2">T</font></span><br/>



<span class="kw">thus </span><a NAME="E1:19"/>
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <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="c2">T</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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">  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font> or <font color="Maroon" title="c3">e</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font>
 ; <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="c3">e</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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">e</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:19_1"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <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">S</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</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> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" onmouseover="rs('yellow_9/D4')" onmouseout="rh()">YELLOW_9:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">A9</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<a NAME="E5:19_1"/>
( <font color="Maroon" title="c4">A</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="waybel32.html#T10" target="_self" onmouseover="rs('waybel32/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E6:19_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">A9</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a>  &amp; <font color="Maroon" title="c5">A9</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="waybel_0.html#T25" onmouseover="rs('waybel_0/T25')" onmouseout="rh()">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47" onmouseover="rs('yellow_9/T47')" onmouseout="rh()">YELLOW_9:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:19_1"/>
<font color="Maroon" title="c3">e</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</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>

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c3">e</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> or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:19"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c3">e</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>
 ; <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">e</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<a NAME="E4:19"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E5:19"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 <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">S</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</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> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" onmouseover="rs('yellow_9/D4')" onmouseout="rh()">YELLOW_9:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">A9</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E7:19"/>
<font color="Maroon" title="c5">A9</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel11.html#D1" onmouseover="rs('waybel11/D1')" onmouseout="rh()">WAYBEL11:def 1</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"> (  not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c1">S</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">A9</font> or  not <font color="Maroon" title="c6">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_2"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">A9</font>
 ; <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">  not <font color="Maroon" title="c6">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A9</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">E</font> = <font color="Maroon" title="c6">D</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E14">A5</font></span></a></span>;<br/>
<a NAME="E3:19_2"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c6">D</font> &amp; (  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c6">D</font> holds <br/><font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D39" onmouseover="rs('waybel_0/D39')" onmouseout="rh()">WAYBEL_0:def 39</a></span>;<br/>
<a NAME="E4:19_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T15" onmouseover="rs('yellow_0/T15')" onmouseout="rh()">YELLOW_0:15</a></span>;<br/>
<a NAME="E5:19_2"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c7">E</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="waybel_0.html#T3" onmouseover="rs('waybel_0/T3')" onmouseout="rh()">WAYBEL_0:3</a></span>;<br/>
<a NAME="E6:19_2"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="yellow_0.html#T26" onmouseover="rs('yellow_0/T26')" onmouseout="rh()">YELLOW_0:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:19_2"/>
 not <font color="Maroon" title="c6">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E1:19_2"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E5:19_2"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="waybel11.html#D1" onmouseover="rs('waybel11/D1')" onmouseout="rh()">WAYBEL11:def 1</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>
<a NAME="E8:19"/><span class="kw">then </span>
( <font color="Maroon" title="c5">A9</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a>  &amp; <font color="Maroon" title="c5">A9</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="waybel_0.html#T25" onmouseover="rs('waybel_0/T25')" onmouseout="rh()">WAYBEL_0:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:19"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel32.html#T10" target="_self" onmouseover="rs('waybel32/T10')" onmouseout="rh()">Th10</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>
