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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="quantal1.html#V14">monotone</a> <a href="quantal1.html#NM1">UnOp</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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:36"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:36"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:36_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:36_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:36_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:36_2_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_1_1"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_1"><b>suppose </b></a><a NAME="E1:36_2_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_2"><b>suppose </b></a><a NAME="E1:36_2_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><a NAME="E2:36_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:36_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:36_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/><b>hence </b><a NAME="E5:36_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="knaster.html#E1:36_2_1"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_3"><b>suppose </b></a><a NAME="E1:36_2_1_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_2_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_4"><b>suppose </b></a><a NAME="E1:36_2_1_1_4"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><a NAME="E2:36_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:36_2_1_1_4"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:36_2_1_1_4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_1_1_4_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  or ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:36_2_1_1_4_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:36_2_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1_4_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_1_1_4_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_2_1_1_4_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:36_2_1_1_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="knaster.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E3:36_2_1_1_4_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36"><i><font color="Green">E27</font></i></a>, <a class="ref" href="knaster.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_2_1_1_4_1_1_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:36_2_1_1_4_1_1_1_1_1_2"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i><a NAME="E4:36_2_1_1_4_1_1_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_2_1_1_4_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="knaster.html#E1:36_2_1_1_4_1_1_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="knaster.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E5:36_2_1_1_4_1_1_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1_4_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:36_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_2_1_1_4_1_1_1_1_1_2_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_1_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:36_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_1_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E6:36_2_1_1_4_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E1:36_2"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E7:36_2_1_1_4_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/>
<a NAME="E8:36_2_1_1_4_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="knaster.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E9:36_2_1_1_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="knaster.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E10:36_2_1_1_4_1_1_1_1_1_2_1"/><i><font color="Green">E41</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_1_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E11:36_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E12:36_2_1_1_4_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:36_2_1_1_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E1:36_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="knaster.html#E10:36_2_1_1_4_1_1_1_1_1_2_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E14:36_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_1_1_1_2_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:36_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E9:36_2_1_1_4_1_1_1_1_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="knaster.html#E13:36_2_1_1_4_1_1_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:36_2_1_1_4_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E4:36_2_1_1_4_1_1_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:36_2_1_1_4_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:36_2_1_1_4_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E4:36_2_1_1_4_1_1_2"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>;<br/><a NAME="E5:36_2_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E6:36_2_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E7:36_2_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/><a NAME="E8:36_2_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="knaster.html#E3:36_2_1_1_4_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="knaster.html#T18" target="_self">Th18</a></i>;<br/><b>hence </b><a NAME="E9:36_2_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="knaster.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:36_2_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_2_1_1_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="knaster.html#E3:36_2_1_1_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/><a NAME="E6:36_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_2"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E7:36_2_1_1_4"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_2_1_1_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="knaster.html#E4:36_2_1_1_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:36_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 ;<br/><b>thus </b><a NAME="E2:36_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:36_3_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:36_3_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E3:36_3_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>6</sub></font> iff ( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_3_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="knaster.html#E1:36_3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="knaster.html#E2:36_3_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_1_1"><b>suppose </b></a><a NAME="E1:36_3_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:36_3_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_3_1_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="knaster.html#E1:36_3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="knaster.html#E3:36_3_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_3_1_1_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:36_3_1_1_1_1_1_2"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i><a NAME="E4:36_3_1_1_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="knaster.html#E1:36_3_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="knaster.html#E3:36_3_1_1_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="knaster.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E5:36_3_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_3_1_1_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="knaster.html#E3:36_3_1_1_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:36_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E4:36_3_1_1_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_1_2"><b>suppose </b></a><a NAME="E1:36_3_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_3_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:36_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="knaster.html#E1:36_3_1"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:36"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="knaster.html#E2:36"><i><font color="Green">E28</font></i></a>, <a class="txt" href="knaster.html#E3:36"><i><font color="Green">E29</font></i></a>, <a class="txt" href="knaster.html#E4:36"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E6:36"/>
<font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>hence </b><a NAME="E7:36"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E5:36"><i><font color="Green">E31</font></i></a></i>;<br/>


</div>
