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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="waybel_9.html#L1">TopRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:51_4_1"/>
( <font color="Maroon">T</font> is <a href="orders_2.html#V2">reflexive</a> &amp; <font color="Maroon">T</font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">T</font> is <a href="pre_topc.html#V2">TopSpace-like</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">T</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V2">reflexive</a>  <a href="waybel_9.html#L1">TopRelStr</a>  ;<br/>
<a NAME="E3:51_4_1"/>
<font color="Maroon">W</font> is <a href="waybel30.html#V2" target="_self">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">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel30.html#D3" target="_self">WAYBEL30:def 3</a><br/>

<b>reconsider </b><font color="Maroon">J</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font></span><a href="tarski.html#K1">}</a></span> as    <a href="yellow13.html#M1">basis</a> of <font color="Maroon">x</font> <b>by </b><i><a class="ref" href="yellow13.html#T21">YELLOW13:21</a></i>;<br/>
<b>take </b>
<font color="Maroon">J</font>
;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W</font>;<br/>

<b>assume </b><a NAME="E2:51_4_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<a NAME="E3:51_4_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:51_4_1_1"/>
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">W</font></span>)</span> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">K</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A</font> as  <a href="yellow_0.html#V7">infs-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E6:51_4_1_1"/>
<font color="Maroon">K</font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 
;<br/>
<b>hence </b><a NAME="E7:51_4_1_1"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A</font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 ;<br/>

<b>thus </b><a NAME="E8:51_4_1_1"/>
<font color="Maroon">A</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:51_4_1"/>
<font color="Maroon">T</font> is <a href="waybel30.html#V2" target="_self">with_compact_semilattices</a>
 ;<br/>


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