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<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> ;<br/>

<b>assume </b><a NAME="E1:51_3_1"/>
<font color="Maroon" title="c1">T</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a>
 ;<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="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a>  ;<br/>
<b>set </b><font color="Maroon" title="c3">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>;<br/>
<div><i><font color="Green" title="E28">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font>;<br/><b>assume </b><a NAME="E1:51_3_1_1"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>
 ;<br/><a NAME="E2:51_3_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="c5">K</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">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#E1:51_3_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:51_3_1_1"/>
<font color="Maroon" title="c5">K</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 ;<br/><b>hence </b><a NAME="E5:51_3_1_1"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:51_3_1"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c2">W</font> is <a href="waybel30.html#V1" title="WAYBEL30:attr.1">with_small_semilattices</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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#D2" target="_self" title="WAYBEL30:def.2">WAYBEL30:def 2</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">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="c4">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="c5">J</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font>;<br/>

<b>assume </b><a NAME="E2:51_3_1_2"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ;<br/>

<b>hence </b><a NAME="E3:51_3_1_2"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c6">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E3:51_3_1"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:51_3_1"/>
<font color="Maroon" title="c2">W</font> is <a href="waybel30.html#V3" title="WAYBEL30:attr.3">with_open_semilattices</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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#D4" target="_self" title="WAYBEL30:def.4">WAYBEL30:def 4</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">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="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c4">x</font> <b>by </b><i><a class="ref" href="yellow13.html#T13" title="YELLOW13:th.13">YELLOW13:13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">J</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font>;<br/>

<b>assume </b><a NAME="E2:51_3_1_3"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ;<br/>

<b>hence </b><a NAME="E3:51_3_1_3"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c6">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E3:51_3_1"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:51_3_1"/>
( <font color="Maroon" title="c1">T</font> is <a href="waybel30.html#V1" title="WAYBEL30:attr.1">with_small_semilattices</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="waybel30.html#V3" title="WAYBEL30:attr.3">with_open_semilattices</a> )
 <b>by </b><i><a class="txt" href="waybel30.html#E4:51_3_1"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/>


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