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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c3">P</font> = <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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c4">a</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="E2:49_1_1"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c4">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#E1:49"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="knaster.html#T24" target="_self" title="KNASTER:th.24">Th24</a></i>;<br/>
<a NAME="E3:49_1_1"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="knaster.html#E2:49_1_1"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<a NAME="E4:49_1_1"/>
<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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c5">P</font> = <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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E6:49_1_1"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c5">P</font> is <a href="knaster.html#V2" title="KNASTER:attr.2">with_suprema</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="knaster.html#D8" target="_self" title="KNASTER:def.8">KNASTER:def 8</a><br/>

<b>assume </b><a NAME="E1:49_1_1_1"/><i><font color="Green" title="E43">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">a</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="E3:49_1_1_1"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">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>consider </b><font color="Maroon" title="c9">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="E5:49_1_1_1"/><i><font color="Green" title="E45">A7</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c9">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#E1:49_1_1_1"><i><font color="Green" title="E43">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c9">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">ab</font> = <font color="Maroon" title="c10">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c11">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>consider </b><font color="Maroon" title="c13">O</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:49_1_1_1"/><i><font color="Green" title="E46">A8</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">O</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</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> &amp; <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</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="c10">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</font> &amp; <font color="Maroon" title="c11">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</font> )
 <b>by </b><i><a class="txt" href="knaster.html#E1:49"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="knaster.html#E3:49_1_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="knaster.html#E5:49_1_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="knaster.html#T35" target="_self" title="KNASTER:th.35">Th35</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</font>;<br/>
<b>take </b>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</font>
;<br/>

<b>thus </b><a NAME="E10:49_1_1_1"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="knaster.html#E9:49_1_1_1"><i><font color="Green" title="E46">A8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:49_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</font> &amp; <font color="Maroon" title="c7">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</font> )
 <b>by </b><i><a class="txt" href="knaster.html#E3:49_1_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="knaster.html#E5:49_1_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="knaster.html#E9:49_1_1_1"><i><font color="Green" title="E46">A8</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c15">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E12:49_1_1_1"/><i><font color="Green" title="E47">A9</font></i>: 
( <font color="Maroon" title="c15">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c6">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">z'</font> &amp; <font color="Maroon" title="c7">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">z'</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">zz</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="E14:49_1_1_1"/><i><font color="Green" title="E48">A10</font></i>: 
( <font color="Maroon" title="c16">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">z'</font> &amp; <font color="Maroon" title="c16">zz</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 ;<br/>
<a NAME="E15:49_1_1_1"/><i><font color="Green" title="E49">A11</font></i>: 
( <font color="Maroon" title="c10">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> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c11">b</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> <font color="Maroon" title="c11">b</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E3:49_1_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="knaster.html#E5:49_1_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>;<br/>
<a NAME="E16:49_1_1_1"/>
( <font color="Maroon" title="c10">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">ab</font> &amp; <font color="Maroon" title="c11">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">ab</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/>
<a NAME="E17:49_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c12">ab</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</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="c12">ab</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E1:49"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</a></i>;<br/>
<a NAME="E18:49_1_1_1"/><b>then </b><i><font color="Green" title="E50">A12</font></i>: 
<font color="Maroon" title="c12">ab</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="c12">ab</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E15:49_1_1_1"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></i>;<br/>
<a NAME="E19:49_1_1_1"/>
<font color="Maroon" title="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">z'</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E3:49_1_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="knaster.html#E5:49_1_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="knaster.html#E12:49_1_1_1"><i><font color="Green" title="E47">A9</font></i></a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></i>;<br/>
<b>hence </b><a NAME="E20:49_1_1_1"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">z'</font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:49"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="knaster.html#E14:49_1_1_1"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="knaster.html#E18:49_1_1_1"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="knaster.html#T37" target="_self" title="KNASTER:th.37">Th37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:49_1_1"/>
<font color="Maroon" title="c5">P</font> is <a href="knaster.html#V3" title="KNASTER:attr.3">with_infima</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="knaster.html#D9" target="_self" title="KNASTER:def.9">KNASTER:def 9</a><br/>

<b>assume </b><a NAME="E1:49_1_1_2"/><i><font color="Green" title="E43">A13</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">a</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="E3:49_1_1_2"/><i><font color="Green" title="E44">A14</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">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>consider </b><font color="Maroon" title="c9">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="E5:49_1_1_2"/><i><font color="Green" title="E45">A15</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c9">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#E1:49_1_1_2"><i><font color="Green" title="E43">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c9">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">ab</font> = <font color="Maroon" title="c10">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>consider </b><font color="Maroon" title="c13">O</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:49_1_1_2"/><i><font color="Green" title="E46">A16</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">O</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</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> &amp; <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</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="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">b</font> )
 <b>by </b><i><a class="txt" href="knaster.html#E1:49"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="knaster.html#E3:49_1_1_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="knaster.html#E5:49_1_1_2"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="knaster.html#T36" target="_self" title="KNASTER:th.36">Th36</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</font>;<br/>
<b>take </b>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</font>
;<br/>

<b>thus </b><a NAME="E10:49_1_1_2"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="knaster.html#E9:49_1_1_2"><i><font color="Green" title="E46">A16</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:49_1_1_2"/>
( <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="knaster.html#E3:49_1_1_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="knaster.html#E5:49_1_1_2"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="knaster.html#E9:49_1_1_2"><i><font color="Green" title="E46">A16</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c15">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E12:49_1_1_2"/><i><font color="Green" title="E47">A17</font></i>: 
( <font color="Maroon" title="c15">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c15">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c15">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">zz</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="E14:49_1_1_2"/><i><font color="Green" title="E48">A18</font></i>: 
( <font color="Maroon" title="c16">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">z'</font> &amp; <font color="Maroon" title="c16">zz</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 ;<br/>
<a NAME="E15:49_1_1_2"/><i><font color="Green" title="E49">A19</font></i>: 
( <font color="Maroon" title="c10">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> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c11">b</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> <font color="Maroon" title="c11">b</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E3:49_1_1_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="knaster.html#E5:49_1_1_2"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>;<br/>
<a NAME="E16:49_1_1_2"/>
( <font color="Maroon" title="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">b</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></i>;<br/>
<a NAME="E17:49_1_1_2"/><b>then </b>
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">ab</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="c10">a</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">ab</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="c11">b</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E1:49"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</a></i>;<br/>
<a NAME="E18:49_1_1_2"/><b>then </b><i><font color="Green" title="E50">A20</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="c12">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">ab</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E15:49_1_1_2"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a></i>;<br/>
<a NAME="E19:49_1_1_2"/>
<font color="Maroon" title="c15">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">ab</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E3:49_1_1_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="knaster.html#E5:49_1_1_2"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="knaster.html#E12:49_1_1_2"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a></i>;<br/>
<b>hence </b><a NAME="E20:49_1_1_2"/>
<font color="Maroon" title="c15">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">O</font> <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c12">ab</font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:49"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="knaster.html#E14:49_1_1_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="knaster.html#E18:49_1_1_2"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="knaster.html#T38" target="_self" title="KNASTER:th.38">Th38</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c5">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="knaster.html#V2" title="KNASTER:attr.2">with_suprema</a> <a href="knaster.html#V3" title="KNASTER:attr.3">with_infima</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="knaster.html#E3:49_1_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="knaster.html#E6:49_1_1"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>
<b>take </b>
 <a href="knaster.html#K7" title="KNASTER:func.7">latt</a> <font color="Maroon" title="c6">P</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">P</font>
;<br/>

<b>thus </b><a NAME="E9:49_1_1"/>
<font color="Maroon" title="c6">P</font> <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> 
 ;<br/>

<b>thus </b><a NAME="E10:49_1_1"/>
 <a href="knaster.html#K7" title="KNASTER:func.7">latt</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K7" title="KNASTER:func.7">latt</a> <font color="Maroon" title="c6">P</font>
 ;<br/>


</div>
