<?xml version="1.0"?>
<div class="add">

<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>set </b><font color="Maroon" title="c3">F</font> =  <a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c4">cF</font> = 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>set </b><font color="Maroon" title="c5">cL</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:53"/><i><font color="Green" title="E43">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/>
<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D18" title="LATTICE3:def.18">LATTICE3:def 18</a><br/>

<b>set </b><font color="Maroon" title="c7">Y</font> = <font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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/>
<a NAME="E2:53"/><i><font color="Green" title="E44">A2</font></i>: 
( <font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">s</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E3:53"/>
<font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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/>

<a NAME="E2:53_1"/><b>then </b>
<font color="Maroon" title="c9">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></i>;<br/>
<a NAME="E3:53_1"/><b>then </b><i><font color="Green" title="E46">A4</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="c9">q</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> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">q'</font> = <font color="Maroon" title="c9">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E5:53_1"/>
<font color="Maroon" title="c10">q'</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#E2:53"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="knaster.html#E1:53_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="knaster.html#T41" target="_self" title="KNASTER:th.41">Th41</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1"/>
<font color="Maroon" title="c9">q</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> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="knaster.html#E3:53_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:53"/><b>then </b><i><font color="Green" title="E45">A5</font></i>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</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> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">O</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:53"/><i><font color="Green" title="E46">A6</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">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="c9">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span> <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#T33" target="_self" title="KNASTER:th.33">Th33</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">p'</font> = <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span> 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="c11">p</font> = <font color="Maroon" title="c10">p'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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#E6:53"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="knaster.html#T41" target="_self" title="KNASTER:th.41">Th41</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">p''</font> = <font color="Maroon" title="c11">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>take </b>
<font color="Maroon" title="c11">p</font>
;<br/>

<b>thus </b><a NAME="E10:53"/>
<font color="Maroon" title="c6">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c11">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:53_2"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c14">q'</font> = <font color="Maroon" title="c13">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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/>
<a NAME="E3:53_2"/>
( <font color="Maroon" title="c13">q</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> &amp; 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="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="knaster.html#T40" target="_self" title="KNASTER:th.40">Th40</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">qL'</font> = <font color="Maroon" title="c13">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E5:53_2"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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#E1:53_2"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:53_2"/><b>then </b><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c15">qL'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></i>;<br/>
<a NAME="E7:53_2"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">p'</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:53"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="knaster.html#T25" target="_self" title="KNASTER:th.25">Th25</a></i>;<br/>
<a NAME="E8:53_2"/><b>then </b>
<font color="Maroon" title="c15">qL'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">p'</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E6:53_2"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E9:53_2"/>
<font color="Maroon" title="c13">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="ref" href="knaster.html#T42" target="_self" title="KNASTER:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c13">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>assume </b><a NAME="E11:53"/><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c13">r</font>
 ;<br/>

<a NAME="E12:53"/>
<font color="Maroon" title="c13">r</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="c14">r'</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="E14:53"/><i><font color="Green" title="E48">A10</font></i>: 
( <font color="Maroon" title="c14">r'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> &amp; <font color="Maroon" title="c14">r'</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#E1:53"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">r''</font> = <font color="Maroon" title="c13">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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/>
<a NAME="E16:53"/>
<font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c14">r'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:53_3"/><i><font color="Green" title="E49">A11</font></i>: 
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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 reconsider </b><font color="Maroon" title="c17">q''</font> = <font color="Maroon" title="c16">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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#E2:53"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">q'</font> = <font color="Maroon" title="c16">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E4:53_3"/>
<font color="Maroon" title="c17">q''</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">r''</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E2:53"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="knaster.html#E11:53"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="knaster.html#E1:53_3"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E5:53_3"/>
<font color="Maroon" title="c16">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">r'</font>
 <b>by </b><i><a class="txt" href="knaster.html#E14:53"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="knaster.html#T42" target="_self" title="KNASTER:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:53"/><b>then </b>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" title="KNASTER:func.8">FixPoints</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">r'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E18:53"/><b>then </b>
<font color="Maroon" title="c10">p'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">r'</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:53"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="knaster.html#E14:53"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="knaster.html#T37" target="_self" title="KNASTER:th.37">Th37</a></i>;<br/>
<b>hence </b><a NAME="E19:53"/>
<font color="Maroon" title="c11">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">r</font>
 <b>by </b><i><a class="txt" href="knaster.html#E14:53"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="knaster.html#T42" target="_self" title="KNASTER:th.42">Th42</a></i>;<br/>


</div>
