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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a> <a href="quantal1.html#NM1" title="QUANTAL1:NM.1">UnOp</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E1:57"/>
<font color="Maroon" title="c3">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b><font color="Maroon" title="c2">f</font>,$1 <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font>;<br/>
<a NAME="E3:57"/>
<font color="Maroon" title="c2">f</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="knaster.html#T16" target="_self" title="KNASTER:th.16">Th16</a></i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="lattices.html#T41" title="LATTICES:th.41">LATTICES:41</a></i>;<br/>
<div><i><font color="Green" title="E48">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O1</font>]
 ;<br/><a NAME="E2:57_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</a></i>;<br/><b>hence </b><a NAME="E3:57_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font>]
 <b>by </b><i><a class="txt" href="knaster.html#E2:57"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="knaster.html#T18" target="_self" title="KNASTER:th.18">Th18</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E49">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:57_2"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c4">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">O1</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>] ) )
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> = <font color="Maroon" title="c2">f</font>,$1 <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c5">L1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E3:57_2"/><i><font color="Green" title="E51">A6</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O1</font> &amp; ( for <font color="Olive" title="b1">O3</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">O1</font> holds <br/><font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">O3</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i><a NAME="E4:57_2"/><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L1</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:57_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="knaster.html#E3:57_2"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="knaster.html#T20" target="_self" title="KNASTER:th.20">Th20</a></i>;<br/><a NAME="E5:57_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L1</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c3">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:57_2_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:57_2_1"/><i><font color="Green" title="E53">A8</font></i>: 
( <font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c7">C</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="knaster.html#E3:57_2_1"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:57_2_1"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">C</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E1:57_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="knaster.html#E3:57_2"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="knaster.html#E3:57_2_1"><i><font color="Green" title="E53">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:57_2_1"/>
<font color="Maroon" title="c6">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="knaster.html#E3:57_2"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="knaster.html#E3:57_2_1"><i><font color="Green" title="E53">A8</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:57_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O1</font>]
 <b>by </b><i><a class="txt" href="knaster.html#E4:57_2"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:57"/>
for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="knaster.html#E4:57"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="knaster.html#E5:57"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="knaster.html#E6:57"><i><font color="Green" title="E49">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:57"/>
 <a href="knaster.html#K9" title="KNASTER:func.9">lfp</a> <font color="Maroon" title="c2">f</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b><font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,$1 <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>
<a NAME="E9:57"/>
<font color="Maroon" title="c2">f</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="knaster.html#T17" target="_self" title="KNASTER:th.17">Th17</a></i>;<br/>
<a NAME="E10:57"/><b>then </b><i><font color="Green" title="E50">A9</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="lattices.html#T45" title="LATTICES:th.45">LATTICES:45</a></i>;<br/>
<div><i><font color="Green" title="E51">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:57_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">O1</font>]
 ;<br/><a NAME="E2:57_3"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</a></i>;<br/><b>hence </b><a NAME="E3:57_3"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font>]
 <b>by </b><i><a class="txt" href="knaster.html#E2:57"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="knaster.html#T19" target="_self" title="KNASTER:th.19">Th19</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E52">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:57_4"/><i><font color="Green" title="E53">A12</font></i>: 
( <font color="Maroon" title="c4">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">O1</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">O1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">O2</font>] ) )
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> = <font color="Maroon" title="c2">f</font>,$1 <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c5">L1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E3:57_4"/><i><font color="Green" title="E54">A13</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O1</font> &amp; ( for <font color="Olive" title="b1">O3</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">O1</font> holds <br/><font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">O3</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i><a NAME="E4:57_4"/><i><font color="Green" title="E55">A14</font></i>: 
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L1</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:57_4"><i><font color="Green" title="E53">A12</font></i></a>, <a class="txt" href="knaster.html#E3:57_4"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="knaster.html#T21" target="_self" title="KNASTER:th.21">Th21</a></i>;<br/><a NAME="E5:57_4"/>
<font color="Maroon" title="c3">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:57_4_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:57_4_1"/><i><font color="Green" title="E56">A15</font></i>: 
( <font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c7">C</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="knaster.html#E3:57_4_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:57_4_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">C</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="knaster.html#E1:57_4"><i><font color="Green" title="E53">A12</font></i></a>, <a class="txt" href="knaster.html#E3:57_4"><i><font color="Green" title="E54">A13</font></i></a>, <a class="txt" href="knaster.html#E3:57_4_1"><i><font color="Green" title="E56">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:57_4_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="txt" href="knaster.html#E3:57_4"><i><font color="Green" title="E54">A13</font></i></a>, <a class="txt" href="knaster.html#E3:57_4_1"><i><font color="Green" title="E56">A15</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:57_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">O1</font>]
 <b>by </b><i><a class="txt" href="knaster.html#E4:57_4"><i><font color="Green" title="E55">A14</font></i></a>, <a class="ref" href="lattice3.html#T40" title="LATTICE3:th.40">LATTICE3:40</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:57"/>
for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">O2</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="knaster.html#E10:57"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="knaster.html#E11:57"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="knaster.html#E12:57"><i><font color="Green" title="E52">A11</font></i></a>);<br/></i>
<b>hence </b><a NAME="E14:57"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="knaster.html#K10" title="KNASTER:func.10">gfp</a> <font color="Maroon" title="c2">f</font>
 ;<br/>


</div>
