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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>consider </b><font color="Maroon">P</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="knaster.html#V2" target="_self">with_suprema</a> <a href="knaster.html#V3" target="_self">with_infima</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:53"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">x</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> </span>}</span>  &amp;  <a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="knaster.html#K7" target="_self">latt</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span>;<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>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E3:53"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>

<b>consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; ( <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> implies <font color="Maroon">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b'</font> ) &amp; ( <font color="Maroon">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b'</font> implies <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> ) )
 <b>by </b><i><a class="ref" href="knaster.html#T4" target="_self">Th4</a>, <a class="txt" href="knaster.html#E5:53"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:53"/>
( <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> iff <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="knaster.html#D3" target="_self">Def3</a></i>;<br/>


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