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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">V</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:33"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#D12" title="WAYBEL11:def.12">WAYBEL11:def 12</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green" title="E26">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:33"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E2:33"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="waybel11.html#T31" title="WAYBEL11:th.31">WAYBEL11:31</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green" title="E28">A5</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T32" title="WAYBEL11:th.32">WAYBEL11:32</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green" title="E29">A6</font></i>: 
(  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a> )
 
;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c3">V</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 ;<br/><b>thus </b><a NAME="E2:33_1"/>
<font color="Maroon" title="c2">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a><br/>

<b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green" title="E31">A8</font></i>: 
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 ;<br/>

<a NAME="E2:33_1_1"/>
( <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="waybel11.html#T12" title="WAYBEL11:th.12">WAYBEL11:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">mdw</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> , <font color="Maroon" title="c7">mdv</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E2:33"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel14.html#E3:33"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="waybel14.html#E5:33"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:33_1_1"/>
( <font color="Maroon" title="c4">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c5">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">w</font> )
 
;<br/>
<a NAME="E5:33_1_1"/><b>then </b>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">w</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E6:33_1_1"/><b>then </b>
( not <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; not <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:33"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E1:33_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:33_1_1"/><b>then </b>
( not <font color="Maroon" title="c3">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">mdv</font> &amp; not <font color="Maroon" title="c3">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">mdw</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E8:33_1_1"/><b>then </b>
not <font color="Maroon" title="c3">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">mdv</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c6">mdw</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:33"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel14.html#E5:33"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="waybel14.html#E6:33"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="waybel14.html#E1:33_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="waybel14.html#T14" target="_self" title="WAYBEL14:th.14">Th14</a></i>;<br/>
<a NAME="E9:33_1_1"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
not <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">mdv</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c6">mdw</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E10:33_1_1"/>
<font color="Maroon" title="c7">mdv</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">mdw</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">mdv</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c6">mdw</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:33"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel14.html#E5:33"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="waybel14.html#E6:33"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="yellow_1.html#T6" title="YELLOW_1:th.6">YELLOW_1:6</a></i>;<br/>
<a NAME="E11:33_1_1"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
not <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">mdv</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">mdw</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E9:33_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<font color="Maroon" title="c7">mdv</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">mdw</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="ref" href="waybel14.html#T3" target="_self" title="WAYBEL14:th.3">Th3</a></i>
;<br/>
<a NAME="E13:33_1_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:33"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E11:33_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/>
<a NAME="E14:33_1_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <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> <span class="p4">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">zz</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:33_1_1"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c8">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E17:33_1_1"/><i><font color="Green" title="E35">A12</font></i>: 
( <font color="Maroon" title="c8">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c8">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">w</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E16:33_1_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">zz</font> = <font color="Maroon" title="c8">zz</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel14.html#E16:33_1_1"><i><font color="Green" title="E34">A11</font></i></a></i>;<br/>
<a NAME="E19:33_1_1"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c9">zz</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">w</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E17:33_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/>
<b>take </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c4">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">w</font>;<br/>

<b>thus </b><a NAME="E20:33_1_1"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E4:33"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="waybel14.html#E17:33_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="waybel14.html#E19:33_1_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/>

<b>thus </b><a NAME="E21:33_1_1"/>
( <font color="Maroon" title="c10">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c10">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">w</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:33_1"/>
<font color="Maroon" title="c2">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E1:33"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E2:33"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel14.html#E3:33"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="waybel11.html#T31" title="WAYBEL11:th.31">WAYBEL11:31</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:33"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c2">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 ;<br/>

<b>assume </b><a NAME="E9:33"/>
not <font color="Maroon" title="c3">V</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">Xx</font>, <font color="Maroon" title="c5">Y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:33"/><i><font color="Green" title="E31">A15</font></i>: 
( <font color="Maroon" title="c3">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">Xx</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">Y</font> &amp; not <font color="Maroon" title="c3">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">Xx</font> &amp; not <font color="Maroon" title="c3">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E2:33"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel14.html#E5:33"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="waybel14.html#E6:33"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="waybel14.html#T14" target="_self" title="WAYBEL14:th.14">Th14</a></i>;<br/>
<a NAME="E12:33"/>
( <font color="Maroon" title="c4">Xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:33"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">Xx'</font> = <font color="Maroon" title="c4">Xx</font>, <font color="Maroon" title="c7">Y'</font> = <font color="Maroon" title="c5">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E14:33"/><i><font color="Green" title="E32">A16</font></i>: 
( <font color="Maroon" title="c6">Xx'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">Y'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:33"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel14.html#E3:33"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="waybel14.html#E5:33"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E15:33"/><b>then </b>
<font color="Maroon" title="c6">Xx'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T37" title="TOPS_1:th.37">TOPS_1:37</a></i>;<br/>
<a NAME="E16:33"/><b>then </b>
<font color="Maroon" title="c4">Xx</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:33"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel14.html#E5:33"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E17:33"/><b>then </b>
<font color="Maroon" title="c4">Xx</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Xx</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T8" title="YELLOW_1:th.8">YELLOW_1:8</a></i>;<br/>
<a NAME="E18:33"/><b>then </b><i><font color="Green" title="E33">A17</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Xx</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E11:33"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E19:33"/><i><font color="Green" title="E34">A18</font></i>: 
( not <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Xx</font> &amp; not <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E11:33"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:33"/><i><font color="Green" title="E35">A19</font></i>: 
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Xx</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E23:33"/><i><font color="Green" title="E36">A20</font></i>: 
( <font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; not <font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E19:33"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E24:33"/><i><font color="Green" title="E37">A21</font></i>: 
( <font color="Maroon" title="c6">Xx'</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Maroon" title="c7">Y'</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E14:33"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c9">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel14.html#E1:33"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E21:33"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="waybel14.html#E23:33"><i><font color="Green" title="E36">A20</font></i></a></i>;<br/>
<div><i><font color="Green" title="E38">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_2"/><i><font color="Green" title="E39">A23</font></i>: 
<font color="Maroon" title="c10">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Xx'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y'</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_2_1"/>
( <font color="Maroon" title="c10">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Xx'</font> or <font color="Maroon" title="c10">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y'</font> )
 </span><b>by </b><i><a class="txt" href="waybel14.html#E1:33_2"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1_1"><b>suppose </b></a><a NAME="E1:33_2_1_1"/><i><font color="Green" title="E40">A24</font></i>: 
<font color="Maroon" title="c10">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Xx'</font>
 ;<br/><div class="add"><a NAME="E2:33_2_1_1"/>
<font color="Maroon" title="c10">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">v</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E3:33_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E21:33"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="waybel14.html#E24:33"><i><font color="Green" title="E37">A21</font></i></a>, <a class="txt" href="waybel14.html#E1:33_2_1_1"><i><font color="Green" title="E40">A24</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1_2"><b>suppose </b></a><a NAME="E1:33_2_1_2"/><i><font color="Green" title="E40">A25</font></i>: 
<font color="Maroon" title="c10">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y'</font>
 ;<br/><div class="add"><a NAME="E2:33_2_1_2"/>
<font color="Maroon" title="c10">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">w</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E3:33_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E23:33"><i><font color="Green" title="E36">A20</font></i></a>, <a class="txt" href="waybel14.html#E24:33"><i><font color="Green" title="E37">A21</font></i></a>, <a class="txt" href="waybel14.html#E1:33_2_1_2"><i><font color="Green" title="E40">A25</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E27:33"/>
<font color="Maroon" title="c10">v</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:33"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E8:33"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="waybel14.html#E21:33"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="waybel14.html#E23:33"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="waybel_0.html#T41" title="WAYBEL_0:th.41">WAYBEL_0:41</a></i>;<br/>
<b>hence </b><a NAME="E28:33"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E1:33"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E18:33"><i><font color="Green" title="E33">A17</font></i></a>, <a class="txt" href="waybel14.html#E26:33"><i><font color="Green" title="E38">A22</font></i></a></i>;<br/>


</div>
