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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>;<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/>





<a NAME="E1:51"/><i><font color="Green" title="E42">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="knaster.html#T39" target="_self" title="KNASTER:th.39">Th39</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:51_1"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</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="txt" href="knaster.html#E1:51"><i><font color="Green" title="E42">A1</font></i></a></i>;<br/><b>thus </b><a NAME="E4:51_1"/>
<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="txt" href="knaster.html#E3:51_1"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:51"/>
<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/>

<b>hence </b><a NAME="E4:51"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="knaster.html#E1:51"><i><font color="Green" title="E42">A1</font></i></a></i>;<br/>


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