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<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="E1:39"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 ;<br/>

<b>let </b><font color="Maroon">O1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">O2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E2:39"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">O1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">O2</font> &amp; not <font color="Maroon">f</font>,<font color="Maroon">O2</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K6" 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#K6" target="_self">-.</a> <font color="Maroon">a</font> )
 ;<br/>

<a NAME="E3:39"/><b>then </b>
<font color="Maroon">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E4:39"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O2</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E5:39"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">O2</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:39"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O1</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="E7:39"/><b>then </b>
<font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:39"/><b>then </b>
<font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span> <a href="knaster.html#K6" 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#K6" target="_self">-.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:39"/><b>then </b>
<font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K6" 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">O1</font></span>)</span> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>
<a NAME="E10:39"/><b>then </b>
<font color="Maroon">f</font>,<font color="Maroon">O1</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <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">O1</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="knaster.html#E1:39"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:39"/>
contradiction
 <b>by </b><i>, <a class="ref" href="abian.html#D4">ABIAN:def 4</a></i>;<br/>


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