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

<span class="kw">let </span><font color="Maroon" title="c1">L1</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L1</font><br/>  for <font color="Olive" title="b2">b</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Olive" title="b1">T</font> holds  <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b3">u</font></span>)</span></span>)</span> where <font color="Olive" title="b3">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">T</font> : <font color="Olive" title="b3">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">b</font> </span> } </span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font> holds  <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b2">u</font></span>)</span></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">b</font> </span> } </span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font> be   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span> } </span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">b2</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span> } </span> ;<br/>
<a NAME="E1:30"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span> } </span>  or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span><br/>

<a NAME="E2:30_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">u</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span> &amp; <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:30_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">b2</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span> } </span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">b1</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> : verum </span> } </span>  as   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T45" onmouseover="rs('waybel11/T45')" onmouseout="rh()">WAYBEL11:45</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E22">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">T</font>  st <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> for <font color="Olive" title="b2">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/> ex <font color="Olive" title="b3">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K32</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>) st <br/>( <font color="Olive" title="b3">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">u</font> &amp; <font color="Olive" title="b3">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  for <font color="Olive" title="b1">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font> holds <br/> ex <font color="Olive" title="b2">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K32</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>) st <br/>( <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b2">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c7">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font> holds <br/> ex <font color="Olive" title="b2">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K32</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>) st <br/>( <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b2">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font> implies  ex <font color="Olive" title="b1">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K32</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>) st <br/>( <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> ) )</span><br/><span class="kw">assume </span><a NAME="E2:30_2"/>
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K32</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>) st <br/>( <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E4:30_2"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c9">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">b1</font>
 <span class="kw">and </span><br/><a NAME="E5:30_2"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">and </span><br/><a NAME="E6:30_2"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/><a NAME="E7:30_2"/>
<font color="Maroon" title="c9">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="yellow_8.html#T10" onmouseover="rs('yellow_8/T10')" onmouseout="rh()">YELLOW_8:10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E9:30_2"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c10">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E10:30_2"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">D</font>
 <span class="kw">and </span><br/><a NAME="E11:30_2"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c10">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/><a NAME="E12:30_2"/>
<font color="Maroon" title="c10">D</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="yellow_8.html#T10" onmouseover="rs('yellow_8/T10')" onmouseout="rh()">YELLOW_8:10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">E</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E14:30_2"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c11">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">b1</font>
 <span class="kw">and </span><br/><a NAME="E15:30_2"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">E</font>
 <span class="kw">and </span><br/><a NAME="E16:30_2"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c11">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E18:30_2"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c12">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_2"><span class="lab"><font color="Green" title="E30">A9</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c13">u</font> =  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">D</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">u</font> &amp; <font color="Maroon" title="c13">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><span class="kw">thus </span><a NAME="E19:30_2"/>
<font color="Maroon" title="c13">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">u</font> &amp; <font color="Maroon" title="c13">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c8">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E22:30_2"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/><a NAME="E23:30_2"/>
<font color="Maroon" title="c12">z</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c14">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E18:30_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="waybel_3.html#T8" onmouseover="rs('waybel_3/T8')" onmouseout="rh()">WAYBEL_3:8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E25:30_2"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">and </span><br/><a NAME="E26:30_2"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c14">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#T52" onmouseover="rs('waybel_4/T52')" onmouseout="rh()">WAYBEL_4:52</a></span>;<br/><a NAME="E27:30_2"/>
(  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">D</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c10">D</font> &amp; <font color="Maroon" title="c16">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c12">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:30_2"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="waybel_3.html#T8" onmouseover="rs('waybel_3/T8')" onmouseout="rh()">WAYBEL_3:8</a>, <a class="ref" href="yellow_0.html#T33" onmouseover="rs('yellow_0/T33')" onmouseout="rh()">YELLOW_0:33</a></span>;<br/><a NAME="E28:30_2"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">D</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:30_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E18:30_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="lattice3.html#D8" onmouseover="rs('lattice3/D8')" onmouseout="rh()">LATTICE3:def 8</a></span>;<br/><a NAME="E29:30_2"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">D</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c14">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:30_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="waybel_3.html#T2" onmouseover="rs('waybel_3/T2')" onmouseout="rh()">WAYBEL_3:2</a></span>;<br/><span class="kw">hence </span><a NAME="E30:30_2"/>
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T8" onmouseover="rs('waybel_3/T8')" onmouseout="rh()">WAYBEL_3:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font></span><br/><a NAME="E31:30_2"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c15">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T11" onmouseover="rs('waybel_3/T11')" onmouseout="rh()">WAYBEL_3:11</a></span>;<br/><a NAME="E32:30_2"/>
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c2">T</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c10">D</font>,<font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" onmouseover="rs('yellow_0/T17')" onmouseout="rh()">YELLOW_0:17</a></span>;<br/><a NAME="E33:30_2"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:30_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E11:30_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E31:30_2"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="yellow_0.html#T35" onmouseover="rs('yellow_0/T35')" onmouseout="rh()">YELLOW_0:35</a></span>;<br/><a NAME="E34:30_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T39" onmouseover="rs('waybel_0/T39')" onmouseout="rh()">WAYBEL_0:39</a></span>;<br/><a NAME="E35:30_2"/><span class="kw">then </span>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">D</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:30_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="waybel_3.html#T12" onmouseover="rs('waybel_3/T12')" onmouseout="rh()">WAYBEL_3:12</a></span>;<br/><span class="kw">hence </span><a NAME="E36:30_2"/>
<font color="Maroon" title="c13">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:30"/>
<font color="Maroon" title="c5">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b2</font> or <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">u</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E3:30_3"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c8">u</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:30_3"/>
<font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font>
 ;<br/>
<a NAME="E5:30_3"/>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c8">u</font></span>)</span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T36" onmouseover="rs('waybel11/T36')" onmouseout="rh()">WAYBEL11:36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:30_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_3"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:30"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span> } </span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="yellow_9.html#T32" onmouseover="rs('yellow_9/T32')" onmouseout="rh()">YELLOW_9:32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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