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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>reconsider </b><font color="Maroon">ze</font> =  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E2:56"/><i><font color="Green">E17</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ze</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T41">LATTICES:41</a></i>;<br/>
<b>then consider </b><font color="Maroon">O</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:56"/><i><font color="Green">E21</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">O</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E5:56"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:56"/>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D6">CARD_1:def 6</a></i>;<br/>
<a NAME="E7:56"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="knaster.html#D3" target="_self">Def3</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E8:56"/><b>then </b>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T60">CARD_3:60</a></i>;<br/>
<a NAME="E9:56"/><b>then </b>
<font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K2">nextcard</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E10:56"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="knaster.html#K5" target="_self">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E11:56"/>
 <a href="knaster.html#K9" target="_self">lfp</a> <font color="Maroon">f</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>

<b>take </b>
<font color="Maroon">O</font>
;<br/>

<b>thus </b><a NAME="E12:56"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">O</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="knaster.html#D3" target="_self">Def3</a></i>;<br/>

<b>thus </b><a NAME="E13:56"/>
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="knaster.html#K9" target="_self">lfp</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


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