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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V3">transitive</a> <a href="waybel19.html#V1" target="_self">lower</a>  <a href="waybel_9.html#L1">TopRelStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>

<b>reconsider </b><font color="Maroon">BB</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<a NAME="E5:11"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:11"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a><br/>

<b>assume </b><a NAME="E8:11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:11"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:11"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:11"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/>
<b>assume </b><a NAME="E14:11"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Q</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:11_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E5:11_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , , , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E6:11_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:11_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:11_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E9:11_1"/><b>then </b>
not <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E10:11_1"/><b>then </b>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E11:11_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E12:11_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E16:11"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>
<b>hence </b><a NAME="E17:11"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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