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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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"> ( <font color="Maroon" title="c1">T</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a> implies <font color="Maroon" title="c1">T</font> is <a href="waybel30.html#V2" title="WAYBEL30:attr.2">with_compact_semilattices</a> )</span><br/>

<b>assume </b><a NAME="E1:51_4_1"/>
( <font color="Maroon" title="c1">T</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</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">T</font> is <a href="waybel30.html#V2" title="WAYBEL30:attr.2">with_compact_semilattices</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">W</font> = <font color="Maroon" title="c1">T</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a>  ;<br/>
<a NAME="E3:51_4_1"/>
<font color="Maroon" title="c2">W</font> is <a href="waybel30.html#V2" title="WAYBEL30:attr.2">with_compact_semilattices</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel30.html#D3" target="_self" title="WAYBEL30:def.3">WAYBEL30:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">J</font> being   <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c3">x</font> st <br/>for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">J</font> holds <br/>(  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b2">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a> &amp; <font color="Olive" title="b2">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">J</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as    <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c3">x</font> <b>by </b><i><a class="ref" href="yellow13.html#T21" title="YELLOW13:th.21">YELLOW13:21</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">J</font>
; <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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">J</font> holds <br/>(  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a> &amp; <font color="Olive" title="b1">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )</span><br/>

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">J</font> implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c5">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a> &amp; <font color="Maroon" title="c5">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> ) )</span><br/>

<b>assume </b><a NAME="E2:51_4_1_1"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c5">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a> &amp; <font color="Maroon" title="c5">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )</span><br/>

<a NAME="E3:51_4_1_1"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:51_4_1_1"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">W</font></span>)</span> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">K</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c5">A</font> as  <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="waybel30.html#E2:51_4_1_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:51_4_1_1"/>
<font color="Maroon" title="c6">K</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 
;<br/>
<b>hence </b><a NAME="E7:51_4_1_1"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c5">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a></span><br/>

<b>thus </b><a NAME="E8:51_4_1_1"/>
<font color="Maroon" title="c5">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E3:51_4_1_1"><i><font color="Green" title="E29">A9</font></i></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>hence </b><a NAME="E4:51_4_1"/>
<font color="Maroon" title="c1">T</font> is <a href="waybel30.html#V2" title="WAYBEL30:attr.2">with_compact_semilattices</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
