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<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="E1:34"/><i><font color="Green">E25</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 <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">a<sub>1</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:34"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="ref" href="knaster.html#T16" target="_self">Th16</a></i>;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><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="E1:34_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><a NAME="E2:34_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</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="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/><a NAME="E3:34_1"/><b>then </b>
<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#E1:34"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/><b>hence </b><a NAME="E4:34_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="ref" href="knaster.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div>
<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/>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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="E1:34_2"/><i><font color="Green">E29</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>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:34_2"/><i><font color="Green">E30</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</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>5</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><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:34_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:34_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="knaster.html#E5:34_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E7:34_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i><a class="txt" href="knaster.html#E3:34_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="knaster.html#E5:34_2"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E8:34_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E3:34_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="knaster.html#E5:34_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:34_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<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>  <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>5</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E7:34_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/><a NAME="E10:34_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="knaster.html#E1:34_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="knaster.html#E5:34_2"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E11:34_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</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>5</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E9:34_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/><b>hence </b><a NAME="E12:34_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="knaster.html#E1:34_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="knaster.html#E3:34_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="knaster.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:34"/>
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#E2:34"><i><font color="Green">E26</font></i></a>, <a class="txt" href="knaster.html#E3:34"><i><font color="Green">E27</font></i></a>, <a class="txt" href="knaster.html#E4:34"><i><font color="Green">E28</font></i></a>);</i><br/>


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