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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> implies for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> is <a href="waybel16.html#V1" title="WAYBEL16:attr.1">completely-irreducible</a> holds <br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">p</font></span>)</span> is  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:35"/>
<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> is <a href="waybel16.html#V1" title="WAYBEL16:attr.1">completely-irreducible</a> holds <br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">p</font></span>)</span> is  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_2.html#V1" title="WAYBEL_2:attr.1">satisfying_MC</a>
 
;<br/>
<b>let </b><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> is <a href="waybel16.html#V1" title="WAYBEL16:attr.1">completely-irreducible</a> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E3:35"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="waybel16.html#V1" title="WAYBEL16:attr.1">completely-irreducible</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E4:35"/>
not  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel16.html#K1" title="WAYBEL16:func.1">Irr</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel16.html#T21" target="_self" title="WAYBEL16:th.21">Th21</a></i>;<br/>
<a NAME="E5:35"/><b>then </b><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="waybel16.html#D4" target="_self" title="WAYBEL16:def.4">Def4</a></i>;<br/>
<a NAME="E6:35"/>
<font color="Maroon" title="c2">p</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a>
 
<b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="waybel16.html#T23" target="_self" title="WAYBEL16:th.23">Th23</a></i>;<br/>
<a NAME="E7:35"/><b>then </b>
<font color="Maroon" title="c2">p</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 
<b>by </b><i><a class="ref" href="waybel_6.html#T27" title="WAYBEL_6:th.27">WAYBEL_6:27</a></i>;<br/>
<a NAME="E8:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E5:35"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="waybel_6.html#T26" title="WAYBEL_6:th.26">WAYBEL_6:26</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">V</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span> as   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E11:35"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">q</font>
 <b>and </b><br/><a NAME="E12:35"/><i><font color="Green" title="E26">A5</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font>
 <b>and </b><br/><a NAME="E13:35"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="waybel16.html#T20" target="_self" title="WAYBEL16:th.20">Th20</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>] <b>means </b>( $1 <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font> &amp; not $1 <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font> );<br/>
<b>reconsider </b><font color="Maroon" title="c5">F</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E15:35"/>
not <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="#E11:35"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="orders_2.html#T30" title="ORDERS_2:th.30">ORDERS_2:30</a></i>;<br/>
<a NAME="E16:35"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><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">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> implies  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font> ) )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:35_1"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>and </b><br/><a NAME="E2:35_1"/><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/><b>take </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c6">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/><b>consider </b><font color="Maroon" title="c9">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:35_1"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>and </b><br/><a NAME="E5:35_1"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c9">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font>
 <b>and </b><br/><a NAME="E6:35_1"/><i><font color="Green" title="E32">A11</font></i>: 
not <font color="Maroon" title="c9">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="#E1:35_1"><i><font color="Green" title="E28">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E8:35_1"/><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>and </b><br/><a NAME="E9:35_1"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c10">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font>
 <b>and </b><br/><a NAME="E10:35_1"/><i><font color="Green" title="E35">A14</font></i>: 
not <font color="Maroon" title="c10">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="#E2:35_1"><i><font color="Green" title="E29">A8</font></i></a></i>;<br/><a NAME="E11:35_1"/><i><font color="Green" title="E36">A15</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E12:35_1"/><b>then </b><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="#E4:35_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="#E5:35_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><a NAME="E13:35_1"/>
not <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:35_1_1"/><i><font color="Green" title="E39">A18</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="#E11:35"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>
<div><i><font color="Green" title="E40">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c11">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">d</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:35_1_1_1"/><i><font color="Green" title="E41">A20</font></i>: 
<font color="Maroon" title="c11">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">y</font>
 <b>and </b><br/><a NAME="E2:35_1_1_1"/><i><font color="Green" title="E42">A21</font></i>: 
<font color="Maroon" title="c11">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">d</font></span><br/><a NAME="E3:35_1_1_1"/>
<font color="Maroon" title="c11">d</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="#E8:35_1"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="#E10:35_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="#E1:35_1_1_1"><i><font color="Green" title="E41">A20</font></i></a>, <a class="txt" href="#E2:35_1_1_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><b>hence </b><a NAME="E4:35_1_1_1"/>
<font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">d</font>
 <b>by </b><i><a class="txt" href="#E12:35"><i><font color="Green" title="E26">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c6">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font>
<b>by </b><i><a class="txt" href="#E4:35_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="#E5:35_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon" title="c6">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="txt" href="#E8:35_1"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="#E9:35_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="#E2:35_1_1"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="#E3:35_1_1"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>
<br/>.= 
<font color="Maroon" title="c8">z</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D3" title="WAYBEL_1:def.3">WAYBEL_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T6" title="WAYBEL_1:th.6">WAYBEL_1:6</a></i>
<br/>.= 
<font color="Maroon" title="c6">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:35_1"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon" title="c6">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>
;<br/>
<b>hence </b><a NAME="E5:35_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:35_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="#E6:35_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E14:35_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E12:35_1"><i><font color="Green" title="E37">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/><b>thus </b><a NAME="E15:35_1"/>
<font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font></span><br/><b>thus </b><a NAME="E16:35_1"/>
<font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="#E16:35"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">F1</font> = <font color="Maroon" title="c6">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <b>by </b><i><a class="ref" href="yellow_7.html#T27" title="YELLOW_7:th.27">YELLOW_7:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b1">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c8">x</font> ) )</span><br/><b>assume </b><a NAME="E1:35_2"/><i><font color="Green" title="E28">A22</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b1">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c8">x</font> )</span><br/><b>take </b><font color="Maroon" title="c9">y</font> =  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c6">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c9">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c8">x</font> )</span><br/><b>thus </b><a NAME="E2:35_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c8">x</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_2_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E2:35_2_1"/><b>then </b><i><font color="Green" title="E29">A23</font></i>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">F1</font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T13" title="YELLOW_7:th.13">YELLOW_7:13</a></i>;<br/>
<a NAME="E3:35_2_1"/><i><font color="Green" title="E30">A24</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E4:35_2_1"/><i><font color="Green" title="E31">A25</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>;<br/>
<b>assume </b><a NAME="E5:35_2_1"/>
not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:35_2_1"/><b>then </b>
<font color="Maroon" title="c9">y</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="c2">p</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E7:35_2_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/>
<b>then </b><i><font color="Green" title="E32">A26</font></i>: <font color="Maroon" title="c2">p</font> = 
<font color="Maroon" title="c2">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T23" title="YELLOW_7:th.23">YELLOW_7:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">F1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:35_2_1"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">F1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="waybel_2.html#D6" title="WAYBEL_2:def.6">WAYBEL_2:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="#E3:35_2_1"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="yellow_7.html#T13" title="YELLOW_7:th.13">YELLOW_7:13</a></i>
<br/>.= 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c6">F</font></span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="#E4:35_2_1"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="waybel16.html#T5" target="_self" title="WAYBEL16:th.5">Th5</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c6">F</font> implies <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">r</font> )</span><br/><b>assume </b><a NAME="E1:35_2_1_2"/>
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c6">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">r</font></span><br/><a NAME="E2:35_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b1">v</font></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow_4.html#T15" title="YELLOW_4:th.15">YELLOW_4:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:35_2_1_2"/><i><font color="Green" title="E33">A27</font></i>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">v</font>
 <b>and </b><br/><a NAME="E5:35_2_1_2"/><i><font color="Green" title="E34">A28</font></i>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/><a NAME="E6:35_2_1_2"/><i><font color="Green" title="E35">A29</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="#E4:35_2_1_2"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E8:35_2_1_2"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font>
 <b>and </b><br/><a NAME="E9:35_2_1_2"/>
<font color="Maroon" title="c12">v1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font>
 <b>and </b><br/><a NAME="E10:35_2_1_2"/><i><font color="Green" title="E37">A31</font></i>: 
not <font color="Maroon" title="c12">v1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="#E5:35_2_1_2"><i><font color="Green" title="E34">A28</font></i></a></i>;<br/><a NAME="E11:35_2_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="#E4:35_2_1_2"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="#E8:35_2_1_2"><i><font color="Green" title="E36">A30</font></i></a>, <a class="txt" href="#E10:35_2_1_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>;<br/><a NAME="E12:35_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="#E6:35_2_1_2"><i><font color="Green" title="E35">A29</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><b>hence </b><a NAME="E13:35_2_1_2"/>
<font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="#E12:35"><i><font color="Green" title="E26">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E10:35_2_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c6">F</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E11:35_2_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E32">A26</font></i></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/>
<b>hence </b><a NAME="E12:35_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:35"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="orders_2.html#T30" title="ORDERS_2:th.30">ORDERS_2:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:35_2"/><b>then </b>
not <font color="Maroon" title="c9">y</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="c2">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:35_2"/><b>then </b><i><font color="Green" title="E29">A32</font></i>: 
not <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</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="E5:35_2"/><i><font color="Green" title="E30">A33</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E6:35_2"/><i><font color="Green" title="E31">A34</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/><a NAME="E7:35_2"/><i><font color="Green" title="E32">A35</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E8:35_2"/>
not <font color="Maroon" title="c8">x</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="c2">p</font>
 <b>by </b><i><a class="txt" href="#E1:35_2"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E9:35_2"/><b>then </b>
not <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</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="E10:35_2"/><b>then </b>
not <font color="Maroon" title="c8">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="#E11:35"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="#E12:35"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="waybel16.html#T28" target="_self" title="WAYBEL16:th.28">Th28</a></i>;<br/><a NAME="E11:35_2"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="#E7:35_2"><i><font color="Green" title="E32">A35</font></i></a></i>;<br/><a NAME="E12:35_2"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="#E6:35_2"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/><a NAME="E13:35_2"/><b>then </b><i><font color="Green" title="E33">A36</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="#E5:35_2"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">D</font> implies  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font> &amp; <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font> ) )</span><br/><b>assume </b><a NAME="E1:35_2_2"/><i><font color="Green" title="E34">A37</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font> &amp; <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font> )</span><br/><a NAME="E2:35_2_2"/>
not <font color="Maroon" title="c10">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_2_2_1"/>
<font color="Maroon" title="c10">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:35_2_2_1"/><b>then </b>
<font color="Maroon" title="c10">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E3:35_2_2_1"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">D</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T3" title="WAYBEL_7:th.3">WAYBEL_7:3</a></i>;<br/>
<a NAME="E4:35_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:35_2_2"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E5:35_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:35_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="waybel_0.html#T34" title="WAYBEL_0:th.34">WAYBEL_0:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c11">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:35_2_2"/><i><font color="Green" title="E35">A38</font></i>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></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/><b>reconsider </b><font color="Maroon" title="c12">d</font> = <font color="Maroon" title="c11">d</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="#E4:35_2_2"><i><font color="Green" title="E35">A38</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c13">d</font> = <font color="Maroon" title="c12">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font> &amp; <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">d</font> )</span><br/><b>thus </b><a NAME="E6:35_2_2"/>
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font>
 <b>by </b><i><a class="txt" href="#E4:35_2_2"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">d</font></span><br/><a NAME="E7:35_2_2"/><i><font color="Green" title="E36">A39</font></i>: 
( <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">d</font> &amp; <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E8:35_2_2"/><i><font color="Green" title="E37">A40</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/><a NAME="E9:35_2_2"/>
not <font color="Maroon" title="c13">d</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="c2">p</font>
 <b>by </b><i><a class="txt" href="#E4:35_2_2"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E10:35_2_2"/><b>then </b>
not <font color="Maroon" title="c13">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</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="E11:35_2_2"/><b>then </b>
not <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="#E11:35"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="#E12:35"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="waybel16.html#T28" target="_self" title="WAYBEL16:th.28">Th28</a></i>;<br/><a NAME="E12:35_2_2"/><b>then </b>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="#E7:35_2_2"><i><font color="Green" title="E36">A39</font></i></a></i>;<br/><a NAME="E13:35_2_2"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="#E8:35_2_2"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E14:35_2_2"/>
<font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">d</font>
 <b>by </b><i><a class="txt" href="#E7:35_2_2"><i><font color="Green" title="E36">A39</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E15:35_2"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#D1" title="WAYBEL_3:def.1">WAYBEL_3:def 1</a></i>;<br/><b>hence </b><a NAME="E16:35_2"/>
<font color="Maroon" title="c9">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="#E13:35_2"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:35"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="waybel_6.html#D1" title="WAYBEL_6:def.1">WAYBEL_6:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
