<?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>, <font color="Maroon">c<sub>5</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="waybel35.html#D6" target="_self">WAYBEL35:def 6</a><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E29</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</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>and </b><br/><a NAME="E3:60"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="E4:60"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<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/>
<a NAME="E5:60"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E2:60"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel35.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E6:60"/><i><font color="Green">E34</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>5</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="E7:60"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E3:60"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_4.html#D4">WAYBEL_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1"/>
(  not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>4</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/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</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>4</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#E4:60"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel35.html#E5:60"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel35.html#E6:60"><i><font color="Green">E34</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="60_1_1"><b>suppose </b></a><a NAME="E1:60_1_1"/>
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>4</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="E3:60_1_1"/><i><font color="Green">E36</font></i>: 
<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>5</sub></font>)
 <b>and </b><br/><a NAME="E4:60_1_1"/><i><font color="Green">E37</font></i>: 
not <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="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>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="E6:60_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E7:60_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#K6" target="_self">SupBelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E8:60_1_1"/><i><font color="Green">E40</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>5</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#E3:60_1_1"><i><font color="Green">E36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><b>thus </b><a NAME="E10:60_1_1"/>
<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#E7:60_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E11:60_1_1"/>
for <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> holds  <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="Olive">b<sub>1</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="E12:60_1_1"/><b>then </b>
 <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> is    <a href="waybel35.html#M1" target="_self">strict_chain</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel35.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E13:60_1_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</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 <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <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#E1:60"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel35.html#E7:60_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel35.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E14:60_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</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#E4:60_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel35.html#E6:60_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_4.html#D4">WAYBEL_4:def 4</a></i>;<br/><b>thus </b><a NAME="E15:60_1_1"/>
<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>5</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#E8:60_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>thus </b><a NAME="E16:60_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel35.html#E4:60_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel35.html#E6:60_1_1"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2"><b>suppose </b></a><a NAME="E1:60_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="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</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>4</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="E3:60_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</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="E4:60_1_2"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E5:60_1_2"/>
<font color="Maroon">c<sub>5</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#E5:60"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel35.html#E6:60"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel35.html#E3:60_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/><b>hence </b><a NAME="E6:60_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>  <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="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E7:60"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel35.html#E4:60_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
