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

<b>let </b><font color="Maroon">c<sub>1</sub></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_3_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V2">anti-discrete</a>  <a href="waybel_9.html#L1">TopRelStr</a>  ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<div><i><font color="Green">E28</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">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:51_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>then </b><i><font color="Green">E29</font></i>: <font color="Maroon">c<sub>4</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>

;<br/><a NAME="E3:51_3_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">c<sub>2</sub></font></span>)</span> is <a href="yellow_0.html#V7">infs-inheriting</a>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>4</sub></font> as  <a href="yellow_0.html#V7">infs-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel30.html#E2:51_3_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E5:51_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 ;<br/><b>hence </b><a NAME="E6:51_3_1_1"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>4</sub></font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:51_3_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel30.html#V1" target="_self">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">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel30.html#D2" target="_self">WAYBEL30:def 2</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> as    <a href="yellow13.html#M1">basis</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="yellow13.html#T21">YELLOW13:21</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:51_3_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>hence </b><a NAME="E3:51_3_1_2"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>6</sub></font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E3:51_3_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:51_3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel30.html#V3" target="_self">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">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel30.html#D4" target="_self">WAYBEL30:def 4</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> as    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="yellow13.html#T13">YELLOW13:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:51_3_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>hence </b><a NAME="E3:51_3_1_3"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>6</sub></font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E3:51_3_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:51_3_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel30.html#V1" target="_self">with_small_semilattices</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel30.html#V3" target="_self">with_open_semilattices</a> )
 <b>by </b><i><a class="txt" href="waybel30.html#E4:51_3_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div>
