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<b>let </b><font color="Maroon">O1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="quantal1.html#V14">monotone</a> <a href="quantal1.html#NM1">UnOp</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><a NAME="E1:40"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> )
 ;<br/>

<b>set </b><font color="Maroon">fa</font> = <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> );<br/>
<a NAME="E2:40"/><i><font color="Green">E21</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="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">O2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">O2</font>]
 ;<br/><b>thus </b><a NAME="E2:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1_1"/>
( <font color="Maroon">O1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font> or <font color="Maroon">O1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1_1"/>
<font color="Maroon">O1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_1_1"/>
<font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_1_2"/>
<font color="Maroon">O1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 ;<br/><div class="add"><a NAME="E2:40_1_1_1_2"/><b>then </b>
<font color="Maroon">O1</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:40_1_1_1_2"/><b>then </b>
<font color="Maroon">O1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><b>hence </b><font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font>,<font color="Maroon">O2</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="abian.html#D3">ABIAN:def 3</a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>
<br/>.= 
<font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="knaster.html#T1" target="_self">Th1</a></i>
;<br/><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">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">O2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">O2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">O2</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">O3</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O3</font>] ) )
 ;<br/><b>thus </b><a NAME="E2:40_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">O2</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>proof </b></a><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">L</font> = <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>;<br/>
<b>consider </b><font color="Maroon">L1</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E2:40_2_1"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font> &amp; ( for <font color="Olive">O3</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">O3</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E3:40_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">O2</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</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">L1</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:40_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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="E1:40_2_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">O3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">O3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O3</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">O3</font> = <font color="Maroon">O3</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_1_1_1"/>
( <font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O3</font> or <font color="Maroon">O3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_2_1_1_1_1"/>
<font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O3</font>
 ;<br/><div class="add"><a NAME="E2:40_2_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font>,<font color="Maroon">O3</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:40_2_1_1_1_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_2_1_1_1_2"/>
<font color="Maroon">O3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O1</font>
 ;<br/><div class="add"><a NAME="E2:40_2_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font>,<font color="Maroon">O3</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="knaster.html#E1:40_2"><i><font color="Green">E26</font></i></a></i>;<br/><b>hence </b><a NAME="E3:40_2_1_1_1_2"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:40_2_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">O2</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>assume </b><a NAME="E6:40_2_1"/>
<font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O2</font>
 ;<br/>

<a NAME="E7:40_2_1"/><b>then </b>
<font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O2</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="txt" href="knaster.html#E1:40_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:40_2_1"/>
<font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<font color="Maroon">O2</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E5:40"/>
for <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O2</font>]
 <b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, , );</i><br/>


</div>
