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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">R</font> be  <a href="waybel35.html#V1" title="WAYBEL35:attr.1">extra-order</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">C</font> be  <a href="waybel35.html#V3" title="WAYBEL35:attr.3">satisfying_SIC</a>  <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>;<br/><b>let </b><font color="Maroon" title="c4">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>







<b>assume </b><a NAME="E1:59"/>
<font color="Maroon" title="c4">d</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>
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c4">d</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="c4">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel35.html#D10" target="_self" title="WAYBEL35:def.10">Def10</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<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>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</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">b</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> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">b</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">p</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E3:59"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</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="E4:59"/><i><font color="Green" title="E31">A3</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="c4">d</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/>
<b>assume </b><a NAME="E5:59"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font>
 ;<br/>

<a NAME="E6:59"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>) <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</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><br/>

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>)
 ;<br/>

<a NAME="E2:59_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</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="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</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> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
;<br/>
<b>hence </b><a NAME="E3:59_1"/>
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:59"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">d</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E3:59"><i><font color="Green" title="E30">A2</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="E8:59"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">d</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E5:59"><i><font color="Green" title="E32">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><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1"/>
(  not  <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="c4">d</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> or  ex <font color="Olive" title="b1">a</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/>(  <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="c4">d</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> &amp; not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font> ) )
 </span><b>by </b><i><a class="txt" href="waybel35.html#E2:59"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="waybel35.html#E4:59"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="waybel35.html#E5:59"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1"><b>suppose </b></a><a NAME="E1:59_2_1_1"/>
not  <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="c4">d</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">a</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="E3:59_2_1_1"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">d</font>
 <b>and </b><br/><a NAME="E4:59_2_1_1"/><i><font color="Green" title="E35">A7</font></i>: 
not <font color="Maroon" title="c6">a</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/><a NAME="E5:59_2_1_1"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59_2_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E6:59_2_1_1"/>
 ex <font color="Olive" title="b1">a</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; not <font color="Olive" title="b1">a</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E4:59_2_1_1"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2"><b>suppose </b></a><a NAME="E1:59_2_1_2"/>
 ex <font color="Olive" title="b1">a</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/>(  <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="c4">d</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> &amp; not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">a</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="E3:59_2_1_2"/><i><font color="Green" title="E34">A8</font></i>: 
 <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="c4">d</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">a</font>
 <b>and </b><br/><a NAME="E4:59_2_1_2"/><i><font color="Green" title="E35">A9</font></i>: 
not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 ;<br/><a NAME="E5:59_2_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E2:59"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="waybel35.html#E4:59"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="waybel35.html#E3:59_2_1_2"><i><font color="Green" title="E34">A8</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="E6:59_2_1_2"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E8:59"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="orders_2.html#T32" title="ORDERS_2:th.32">ORDERS_2:32</a></i>;<br/><b>hence </b><a NAME="E7:59_2_1_2"/>
 ex <font color="Olive" title="b1">a</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; not <font color="Olive" title="b1">a</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E4:59_2_1_2"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">cc</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="E11:59"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c6">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E12:59"/><i><font color="Green" title="E35">A11</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">cc</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>and </b><br/><a NAME="E13:59"/><i><font color="Green" title="E36">A12</font></i>: 
not <font color="Maroon" title="c6">cc</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font>
 ;<br/>
<a NAME="E14:59"/><i><font color="Green" title="E37">A13</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">cc</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_3"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">cc</font>,<font color="Maroon" title="c6">cc</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> or  not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">cc</font>,<font color="Maroon" title="c6">cc</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1"><b>suppose </b></a><a NAME="E1:59_3_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">cc</font>,<font color="Maroon" title="c6">cc</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ;<br/><div class="add"><a NAME="E2:59_3_1"/><b>then </b>
<font color="Maroon" title="c6">cc</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">cc</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="waybel35.html#E14:59"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="waybel35.html#T21" target="_self" title="WAYBEL35:th.21">Th21</a></i>;<br/><a NAME="E3:59_3_1"/><b>then </b>
<font color="Maroon" title="c6">cc</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/><a NAME="E4:59_3_1"/><b>then </b>
<font color="Maroon" title="c6">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>)
 <b>by </b><i><a class="txt" href="waybel35.html#E12:59"><i><font color="Green" title="E35">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:59_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="waybel35.html#E13:59"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_2"><b>suppose </b></a><a NAME="E1:59_3_2"/><i><font color="Green" title="E38">A14</font></i>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">cc</font>,<font color="Maroon" title="c6">cc</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ;<br/><div class="add"><a NAME="E2:59_3_2"/>
 ex <font color="Olive" title="b1">cs</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">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">cs</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_3_2_1_1"/>
(  not  <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="c4">d</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">cc</font> or  ex <font color="Olive" title="b1">a</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/>(  <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="c4">d</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> &amp; not <font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font> ) )
 </span><b>by </b><i><a class="txt" href="waybel35.html#E2:59"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="waybel35.html#E4:59"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="waybel35.html#E12:59"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="waybel35.html#E1:59_3_2"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_3_2_1_1_1"/>
not  <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="c4">d</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">cc</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">cs</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="E3:59_3_2_1_1_1"/><i><font color="Green" title="E39">A15</font></i>: 
<font color="Maroon" title="c7">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">d</font>
 <b>and </b><br/><a NAME="E4:59_3_2_1_1_1"/><i><font color="Green" title="E40">A16</font></i>: 
not <font color="Maroon" title="c7">cs</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">cc</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">cs</font>
;<br/><b>thus </b><a NAME="E5:59_3_2_1_1_1"/><i><font color="Green" title="E41">A17</font></i>: 
<font color="Maroon" title="c7">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59_3_2_1_1_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></i>;<br/><a NAME="E6:59_3_2_1_1_1"/>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">cs</font>,<font color="Maroon" title="c6">cc</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E4:59_3_2_1_1_1"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></i>;<br/><a NAME="E7:59_3_2_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">cc</font>,<font color="Maroon" title="c7">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="waybel35.html#E4:59_3_2_1_1_1"><i><font color="Green" title="E40">A16</font></i></a>, <a class="txt" href="waybel35.html#E5:59_3_2_1_1_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></i>;<br/><a NAME="E8:59_3_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">cs</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></i>;<br/><b>hence </b><a NAME="E9:59_3_2_1_1_1"/>
<font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c7">cs</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E4:59_3_2_1_1_1"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><b>thus </b><a NAME="E10:59_3_2_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">cs</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59_3_2_1_1_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_3_2_1_1_2"/><i><font color="Green" title="E39">A18</font></i>: 
 ex <font color="Olive" title="b1">a</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/>(  <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="c4">d</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> &amp; not <font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font> )
 ;<br/><div class="add"><a NAME="E2:59_3_2_1_1_2"/>
<font color="Maroon" title="c6">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">d</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="waybel35.html#E12:59"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E3:59_3_2_1_1_2"/>
 ex <font color="Olive" title="b1">cs</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">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">cs</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E1:59_3_2_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c7">cs</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="E4:59_3_2"/><i><font color="Green" title="E39">A19</font></i>: 
<font color="Maroon" title="c7">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E5:59_3_2"/><i><font color="Green" title="E40">A20</font></i>: 
<font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c7">cs</font>
 <b>and </b><br/><a NAME="E6:59_3_2"/><i><font color="Green" title="E41">A21</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">cs</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ;<br/><b>consider </b><font color="Maroon" title="c8">y</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="E8:59_3_2"/><i><font color="Green" title="E42">A22</font></i>: 
<font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c8">y</font>
 <b>and </b><br/><a NAME="E9:59_3_2"/><i><font color="Green" title="E43">A23</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>and </b><br/><a NAME="E10:59_3_2"/><i><font color="Green" title="E44">A24</font></i>: 
<font color="Maroon" title="c8">y</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="c8">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="waybel35.html#E4:59_3_2"><i><font color="Green" title="E39">A19</font></i></a>, <a class="txt" href="waybel35.html#E5:59_3_2"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="waybel35.html#T22" target="_self" title="WAYBEL35:th.22">Th22</a></i>;<br/><a NAME="E11:59_3_2"/><i><font color="Green" title="E45">A25</font></i>: 
<font color="Maroon" title="c8">y</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="txt" href="waybel35.html#E10:59_3_2"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="waybel35.html#D10" target="_self" title="WAYBEL35:def.10">Def10</a></i>;<br/><a NAME="E12:59_3_2"/><i><font color="Green" title="E46">A26</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">d</font>
 ;<br/><a NAME="E13:59_3_2"/>
<font color="Maroon" title="c8">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">cs</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E9:59_3_2"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></i>;<br/><a NAME="E14:59_3_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c4">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E6:59_3_2"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="waybel35.html#E12:59_3_2"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="waybel_4.html#D5" title="WAYBEL_4:def.5">WAYBEL_4:def 5</a></i>;<br/><a NAME="E15:59_3_2"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>)
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59_3_2"><i><font color="Green" title="E45">A25</font></i></a></i>;<br/><a NAME="E16:59_3_2"/><b>then </b>
<font color="Maroon" title="c8">y</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></i>;<br/><a NAME="E17:59_3_2"/><b>then </b>
<font color="Maroon" title="c6">cc</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">d</font>),<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E8:59_3_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="orders_2.html#T32" title="ORDERS_2:th.32">ORDERS_2:32</a></i>;<br/><b>hence </b><a NAME="E18:59_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel35.html#E13:59"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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