<?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="lattice3.html#V3" title="LATTICE3:attr.3">complete</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="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/> for <font color="Olive" title="b2">C</font> being <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="Olive" title="b1">R</font> holds  <font color="Olive" title="b1">R</font> <a href="waybel35.html#R1" title="WAYBEL35:pred.1">satisfies_SIC_on</a>  <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></span><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>; <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="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> holds  <font color="Maroon" title="c2">R</font> <a href="waybel35.html#R1" title="WAYBEL35:pred.1">satisfies_SIC_on</a>  <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></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">R</font> <a href="waybel35.html#R1" title="WAYBEL35:pred.1">satisfies_SIC_on</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></span><br/><b>let </b><font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</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="waybel35.html#D6" target="_self" title="WAYBEL35:def.6">WAYBEL35:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">c</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; <font color="Maroon" title="c5">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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> implies  ex <font color="Olive" title="b1">y</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">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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> ) )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c4">c</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>and </b><br/><a NAME="E2:58"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c5">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>
 <b>and </b><br/><a NAME="E3:58"/><i><font color="Green" title="E27">A3</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">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="E4:58"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">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">y</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">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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> )</span><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/>
<a NAME="E5:58"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c5">d</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">d</font>),<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="waybel35.html#T27" target="_self" title="WAYBEL35:th.27">Th27</a></i>;<br/>
<a NAME="E6:58"/><i><font color="Green" title="E30">A6</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="c5">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="E7:58"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">d</font>
 
<b>by </b><i><a class="txt" href="#E3:58"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1"/>
(  not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">d</font>) <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">c</font> or  ex <font color="Olive" title="b1">g</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">d</font>) <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">g</font> &amp; not <font color="Maroon" title="c4">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">g</font> ) )
 </span><b>by </b><i><a class="txt" href="#E4:58"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E5:58"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="#E6:58"><i><font color="Green" title="E30">A6</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="58_1_1"><b>suppose </b></a><a NAME="E1:58_1_1"/>
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">d</font>) <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">c</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="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">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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">g</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="E3:58_1_1"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">d</font>)
 <b>and </b><br/><a NAME="E4:58_1_1"/><i><font color="Green" title="E33">A9</font></i>: 
not <font color="Maroon" title="c6">g</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">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="E6:58_1_1"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>and </b><br/><a NAME="E7:58_1_1"/><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c7">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>and </b><br/><a NAME="E8:58_1_1"/><i><font color="Green" title="E36">A12</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">y</font>,<font color="Maroon" title="c5">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="#E3:58_1_1"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>take </b>
<font color="Maroon" title="c8">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">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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> )</span><br/><b>thus </b><a NAME="E10:58_1_1"/>
<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="#E7:58_1_1"><i><font color="Green" title="E35">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> )</span><br/><a NAME="E11:58_1_1"/>
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>
 <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="E12:58_1_1"/><b>then </b>
 <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#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="ref" href="waybel35.html#T25" target="_self" title="WAYBEL35:th.25">Th25</a></i>;<br/><a NAME="E13:58_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="c4">c</font>,<font color="Maroon" title="c8">y</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 <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> or <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">c</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="#E1:58"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E7:58_1_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E14:58_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c8">y</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="#E4:58_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="#E6:58_1_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> )</span><br/><b>thus </b><a NAME="E15:58_1_1"/>
<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="c5">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="#E8:58_1_1"><i><font color="Green" title="E36">A12</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="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font></span><br/><b>thus </b><a NAME="E16:58_1_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="#E4:58_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="#E6:58_1_1"><i><font color="Green" title="E34">A10</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2"><b>suppose </b></a><a NAME="E1:58_1_2"/>
 ex <font color="Olive" title="b1">g</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">d</font>) <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">g</font> &amp; not <font color="Maroon" title="c4">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">g</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="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">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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">g</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="E3:58_1_2"/><i><font color="Green" title="E32">A13</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">d</font>) <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">g</font>
 <b>and </b><br/><a NAME="E4:58_1_2"/><i><font color="Green" title="E33">A14</font></i>: 
not <font color="Maroon" title="c4">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g</font>
 ;<br/><a NAME="E5:58_1_2"/>
<font color="Maroon" title="c5">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="#E5:58"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="#E6:58"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E3:58_1_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/><b>hence </b><a NAME="E6:58_1_2"/>
 ex <font color="Olive" title="b1">y</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">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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="#E7:58"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E4:58_1_2"><i><font color="Green" title="E33">A14</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></div><b>end;</b></div>

</div>
