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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">R</font> being <a href="waybel_4.html#V1" title="WAYBEL_4:attr.1">auxiliary(i)</a> <a href="waybel_4.html#V2" title="WAYBEL_4:attr.2">auxiliary(ii)</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st ( for <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">c</font>,<font color="Maroon" title="c1">L</font> ) holds <br/> <a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">C</font> is <a href="waybel35.html#V4" title="WAYBEL35:attr.4">sup-closed</a></span><br/><b>let </b><font color="Maroon" title="c2">R</font> be  <a href="waybel_4.html#V1" title="WAYBEL_4:attr.1">auxiliary(i)</a> <a href="waybel_4.html#V2" title="WAYBEL_4:attr.2">auxiliary(ii)</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</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"> for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st ( for <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">c</font>,<font color="Maroon" title="c1">L</font> ) holds <br/> <a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">C</font> is <a href="waybel35.html#V4" title="WAYBEL35:attr.4">sup-closed</a></span><br/><b>let </b><font color="Maroon" title="c3">C</font> be   <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"> ( ( for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">L</font> ) implies  <a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font> is <a href="waybel35.html#V4" title="WAYBEL35:attr.4">sup-closed</a> )</span><br/>





<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E24">A1</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Olive" title="b1">c</font>,<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">  <a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font> is <a href="waybel35.html#V4" title="WAYBEL35:attr.4">sup-closed</a></span><br/>

<b>let </b><font color="Maroon" title="c4">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel35.html#D9" target="_self" title="WAYBEL35:def.9">WAYBEL35:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> implies  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> )</span><br/>

<b>set </b><font color="Maroon" title="c5">s</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E2:56"/><i><font color="Green" title="E25">A2</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>assume </b><a NAME="E3:56"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<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">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span></span><br/>

<a NAME="E4:56"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<a NAME="E5:56"/>
 <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel35.html#T19" target="_self" title="WAYBEL35:th.19">Th19</a></i>;<br/>
<a NAME="E6:56"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E4:56"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E7:56"/>
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</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="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:56_1"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:56_1"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c6">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel35.html#D10" target="_self" title="WAYBEL35:def.10">Def10</a></i>;<br/>
<a NAME="E3:56_1"/><i><font color="Green" title="E31">A8</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<a NAME="E4:56_1"/>
 <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c6">x</font> <a href="lattice7.html#R1" title="LATTICE7:pred.1">c=</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E1:56_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="waybel35.html#T20" target="_self" title="WAYBEL35:th.20">Th20</a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></i>;<br/>
<b>hence </b><a NAME="E5:56_1"/>
<font color="Maroon" title="c6">x</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="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:56"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="#E2:56_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="#E3:56_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_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>
<a NAME="E8:56"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</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="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E9:56"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:56"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>
<a NAME="E10:56"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="waybel35.html#D10" target="_self" title="WAYBEL35:def.10">Def10</a></i>;<br/>
<b>thus </b><a NAME="E11:56"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel35.html#K6" title="WAYBEL35:func.6">SupBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:56"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="#E3:56"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E10:56"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="yellow_0.html#T65" title="YELLOW_0:th.65">YELLOW_0:65</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>
