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<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S2</font> #) &amp; <font color="Maroon" title="c1">S1</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> implies <font color="Maroon" title="c2">S2</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> )</span><br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E12">A1</font></i>: 
 <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S2</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="c1">S1</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> or <font color="Maroon" title="c2">S2</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> )</span><br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green" title="E13">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="c1">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S1</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">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S2</font> #)
 
;<br/>
<b>assume </b><a NAME="E3:21"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c1">S1</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"> <font color="Maroon" title="c2">S2</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a></span><br/>

<b>let </b><font color="Maroon" title="c3">T</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</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">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or ( <font color="Maroon" title="c3">T</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> ) ) &amp; (  not <font color="Maroon" title="c3">T</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> or  not <font color="Maroon" title="c3">T</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> or <font color="Maroon" title="c3">T</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">T1</font> = <font color="Maroon" title="c3">T</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:21"/>
( <font color="Maroon" title="c3">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies ( <font color="Maroon" title="c3">T</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">T</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">T</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> or  not <font color="Maroon" title="c3">T</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> or <font color="Maroon" title="c3">T</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="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon" title="c3">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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">T</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )</span><br/>

<a NAME="E2:21_1"/><b>then </b>
<font color="Maroon" title="c3">T</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">S2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:21_1"/><b>then </b>
<font color="Maroon" title="c4">T1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:21_1"/><b>then </b>
( <font color="Maroon" title="c4">T1</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c4">T1</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
( <font color="Maroon" title="c3">T</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47" title="YELLOW_9:th.47">YELLOW_9:47</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>thus </b><a NAME="E6:21"/>
( <font color="Maroon" title="c3">T</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> implies <font color="Maroon" title="c3">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/>
( <font color="Maroon" title="c3">T</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c3">T</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">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<a NAME="E2:21_2"/><b>then </b>
( <font color="Maroon" title="c4">T1</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c4">T1</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 
<b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47" title="YELLOW_9:th.47">YELLOW_9:47</a></i>;<br/>
<a NAME="E3:21_2"/><b>then </b>
<font color="Maroon" title="c4">T1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<a NAME="E4:21_2"/><b>then </b>
<font color="Maroon" title="c4">T1</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="c1">S1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:21_2"/>
<font color="Maroon" title="c3">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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>


</div>
