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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel35.html#V1" target="_self">extra-order</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel35.html#V3" target="_self">satisfying_SIC</a>  <a href="waybel35.html#M1" target="_self">strict_chain</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:59"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel35.html#D10" target="_self">Def10</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:59"/><i><font color="Green">E30</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E4:59"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>assume </b><a NAME="E5:59"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E6:59"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>) <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 ;<br/>

<a NAME="E2:59_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:59_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D4">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">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E3:59"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E8:59"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E5:59"><i><font color="Green">E32</font></i></a>, <a class="ref" href="orders_2.html#D10">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" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; not  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="waybel35.html#E2:59"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel35.html#E4:59"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel35.html#E5:59"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_0.html#D9">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" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:59_2_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:59_2_1_1"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E5:59_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self">Th18</a></i>;<br/><b>hence </b><a NAME="E6:59_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E4:59_2_1_1"><i><font color="Green">E35</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">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; not  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:59_2_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:59_2_1_2"/><i><font color="Green">E35</font></i>: 
not  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E5:59_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E2:59"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel35.html#E4:59"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel35.html#E3:59_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/><a NAME="E6:59_2_1_2"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E8:59"><i><font color="Green">E33</font></i></a>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/><b>hence </b><a NAME="E7:59_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E4:59_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="orders_2.html#D10">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">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:59"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E12:59"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E13:59"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E14:59"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">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">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or  not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel35.html#E14:59"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel35.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:59_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="waybel35.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:59_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="txt" href="waybel35.html#E12:59"><i><font color="Green">E35</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">E30</font></i></a>, <a class="txt" href="waybel35.html#E13:59"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow_4.html#T1">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">E38</font></i>: 
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_3_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>6</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="waybel35.html#E2:59"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel35.html#E4:59"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel35.html#E12:59"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel35.html#E1:59_3_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow_0.html#D9">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" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:59_3_2_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:59_3_2_1_1_1"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/><b>thus </b><a NAME="E5:59_3_2_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59_3_2_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E6:59_3_2_1_1_1"/>
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E4:59_3_2_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_4.html#D4">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">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel35.html#E4:59_3_2_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel35.html#E5:59_3_2_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel35.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E8:59_3_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D4">WAYBEL_4:def 4</a></i>;<br/><b>hence </b><a NAME="E9:59_3_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E4:59_3_2_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="orders_2.html#D10">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">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59_3_2_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self">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">E39</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:59_3_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel35.html#E12:59"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self">Th18</a></i>;<br/><b>hence </b><a NAME="E3:59_3_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E1:59_3_2_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="lattice3.html#D9">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">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:59_3_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:59_3_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E6:59_3_2"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:59_3_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E9:59_3_2"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E10:59_3_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel35.html#E4:59_3_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel35.html#E5:59_3_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel35.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E11:59_3_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E10:59_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel35.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E12:59_3_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E13:59_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E9:59_3_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_4.html#D4">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">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E6:59_3_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel35.html#E12:59_3_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_4.html#D5">WAYBEL_4:def 5</a></i>;<br/><a NAME="E15:59_3_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="txt" href="waybel35.html#E11:59_3_2"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E16:59_3_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:59"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/><a NAME="E17:59_3_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2">&lt;</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E8:59_3_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="orders_2.html#T32">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">E36</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
