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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="waybel_0.html#V24">up-complete</a>  <a href="waybel_9.html#L1">TopRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E13</font></i>: 
 <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #)
 ;<br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 
;<br/>
<b>assume </b><a NAME="E3:21"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel11.html#V4">Scott</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel29.html#E1:21"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:21"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> implies ( <font color="Maroon">c<sub>3</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V13">upper</a> ) )
 <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">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:21_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:21"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:21_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E3:21"><i><font color="Green">E15</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V13">upper</a> )
 <b>by </b><i><a class="txt" href="waybel29.html#E2:21"><i><font color="Green">E14</font></i></a>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47">YELLOW_9:47</a></i>;<br/>


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

<b>thus </b><a NAME="E6:21"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V13">upper</a> implies <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <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">c<sub>3</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V13">upper</a> )
 ;<br/>

<a NAME="E2:21_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E2:21"><i><font color="Green">E14</font></i></a>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47">YELLOW_9:47</a></i>;<br/>
<a NAME="E3:21_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel29.html#E3:21"><i><font color="Green">E15</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E4:21_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:21_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel29.html#E1:21"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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


</div>
