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<span class="kw">let </span><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_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>   <a href="waybel32.html#V1" title="WAYBEL32:attr.1">upper</a>  <a href="waybel19.html#NM1" title="WAYBEL19:NM.1">TopPoset</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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">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/><font color="Olive" title="b1">A</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">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="c2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c2">A</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:6"/>
<font color="Maroon" title="c2">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"> <font color="Maroon" title="c2">A</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> </span><br/>

<a NAME="E2:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">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#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></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="c1">T</font> : verum </span> } </span>  as   <a href="cantor_1.html#NM2" title="CANTOR_1:NM.2">prebasis</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel32.html#D1" target="_self" onmouseover="rs('waybel32/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">F</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E5:6"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">BB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cantor_1.html#D4" onmouseover="rs('cantor_1/D4')" onmouseout="rh()">CANTOR_1:def 4</a></span>;<br/>
<a NAME="E6:6"/>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cantor_1.html#D2" onmouseover="rs('cantor_1/D2')" onmouseout="rh()">CANTOR_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">Y</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E8:6"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">and </span><br/><a NAME="E9:6"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="cantor_1.html#D1" onmouseover="rs('cantor_1/D1')" onmouseout="rh()">CANTOR_1:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D20" onmouseover="rs('waybel_0/D20')" onmouseout="rh()">WAYBEL_0:def 20</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="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E10:6"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">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"> (  not <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E12:6"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">and </span><br/><a NAME="E13:6"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon" title="c8">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E14:6"/>
<font color="Maroon" title="c8">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E13:6"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">X</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E16:6"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c9">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">BB</font>
 <span class="kw">and </span><br/><a NAME="E17:6"/>
<font color="Maroon" title="c9">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">and </span><br/><a NAME="E18:6"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="cantor_1.html#D3" onmouseover="rs('cantor_1/D3')" onmouseout="rh()">CANTOR_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E19:6"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</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">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1"><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">Q</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Q</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">Q</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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"> ( <font color="Maroon" title="c10">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> implies <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Q</font> )</span><br/><span class="kw">assume </span><a NAME="E1:6_1"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</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">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Q</font></span><br/><a NAME="E2:6_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">BB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:6_1"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">z</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/><a NAME="E5:6_1"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:6"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E18:6"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="setfam_1.html#T43" onmouseover="rs('setfam_1/T43')" onmouseout="rh()">SETFAM_1:43</a></span>;<br/><a NAME="E6:6_1"/>
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">z</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>;<br/><a NAME="E7:6_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><a NAME="E8:6_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/><a NAME="E9:6_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:6"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="orders_2.html#T3" onmouseover="rs('orders_2/T3')" onmouseout="rh()">ORDERS_2:3</a></span>;<br/><a NAME="E10:6_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/><span class="kw">hence </span><a NAME="E11:6_1"/>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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="E21:6"/><span class="kw">then </span>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:6"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="setfam_1.html#T43" onmouseover="rs('setfam_1/T43')" onmouseout="rh()">SETFAM_1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:6"/>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E13:6"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</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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