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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="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">a</font> holds <br/> <a href="knaster.html#K9" title="KNASTER:func.9">lfp</a> <font color="Olive" title="b1">f</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</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="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> holds <br/> <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="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="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="c3">a</font> implies  <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> )</span><br/>





<b>assume </b><a NAME="E1:58"/><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="lattices.html#R3" title="LATTICES:pred.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">  <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></span><br/>

<b>then consider </b><font color="Maroon" title="c4">O</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:58"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E4:58"/><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</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>
 <b>by </b><i><a class="ref" href="knaster.html#T34" target="_self" title="KNASTER:th.34">Th34</a></i>;<br/>
<a NAME="E5:58"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</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#E1:58"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="knaster.html#T26" target="_self" title="KNASTER:th.26">Th26</a></i>;<br/>
<a NAME="E6:58"/>
 <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="c2">f</font>,<font color="Maroon" title="c4">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:58"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="knaster.html#T46" target="_self" title="KNASTER:th.46">Th46</a></i>;<br/>
<b>hence </b><a NAME="E7:58"/>
 <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>
 <b>by </b><i><a class="txt" href="knaster.html#E5:58"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</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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