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<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <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/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> holds <br/>for <font color="Olive" title="b3">O1</font>, <font color="Olive" title="b4">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b3">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b4">O2</font> &amp; not <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Olive" title="b2">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Olive" title="b2">a</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">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> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">O1</font>, <font color="Olive" title="b3">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b3">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Olive" title="b1">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Olive" title="b1">a</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> <font color="Maroon" title="c3">a</font> implies for <font color="Olive" title="b1">O1</font>, <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <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> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> )</span><br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E28">A1</font></i>: 
<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> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">O1</font>, <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <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> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/>

<b>let </b><font color="Maroon" title="c4">O1</font>, <font color="Maroon" title="c5">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <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> implies <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> )</span><br/>



<b>assume </b><a NAME="E2:37"/><i><font color="Green" title="E29">A2</font></i>: 
( <font color="Maroon" title="c4">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">O2</font> &amp; not <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <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> &amp; <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:37"/><b>then </b>
<font color="Maroon" title="c4">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O2</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<a NAME="E4:37"/><b>then </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">O2</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E5:37"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="knaster.html#T27" target="_self" title="KNASTER:th.27">Th27</a></i>;<br/>
<a NAME="E6:37"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">O1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>;<br/>
<a NAME="E7:37"/><b>then </b>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="knaster.html#T27" target="_self" title="KNASTER:th.27">Th27</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:37"/><b>then </b>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O1</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E2:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="knaster.html#E5:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></i>;<br/>
<a NAME="E9:37"/><b>then </b>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="knaster.html#T18" target="_self" title="KNASTER:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E10:37"/>
contradiction
 <b>by </b><i><a class="txt" href="knaster.html#E2:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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