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

<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>set </b><font color="Maroon">F</font> =  <a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">cF</font> = the <a href="struct_0.html#U1">carrier</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>set </b><font color="Maroon">cL</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E1:54"/><i><font color="Green">E17</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> <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> 
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D18">LATTICE3:def 18</a><br/>

<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</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/>
<a NAME="E2:54"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font>;<br/>
<a NAME="E3:54"/>
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</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/>

<a NAME="E2:54_1"/><b>then </b>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<a NAME="E3:54_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E5:54_1"/>
<font color="Maroon">q'</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="knaster.html#D3" target="_self">Def3</a>, </i>;<br/>
<b>hence </b><a NAME="E6:54_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:54"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</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="E6:54"/><i><font color="Green">E25</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> &amp; <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="lattice3.html#K15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<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/>
<b>reconsider </b><font color="Maroon">p'</font> = <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="lattice3.html#K15">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">p''</font> = <font color="Maroon">p</font> as   <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>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E10:54"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:54_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <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/>
<a NAME="E3:54_2"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">qL'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E5:54_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:54_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">qL'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<a NAME="E7:54_2"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="knaster.html#D3" target="_self">Def3</a>, <a class="txt" href="knaster.html#E4:54"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E8:54_2"/><b>then </b>
<font color="Maroon">qL'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E9:54_2"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</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>assume </b><a NAME="E11:54"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E12:54"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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>then consider </b><font color="Maroon">r'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:54"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">r'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r'</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="knaster.html#T4" target="_self">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r''</font> = <font color="Maroon">r</font> as   <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/>
<a NAME="E16:54"/>
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</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>then reconsider </b><font color="Maroon">q''</font> = <font color="Maroon">q</font> as   <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> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E4:54_3"/>
<font color="Maroon">q''</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r''</font>
 
<b>by </b><i>, , , <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E5:54_3"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:54"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="knaster.html#K8" target="_self">FixPoints</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E18:54"/><b>then </b>
<font color="Maroon">p'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r'</font>
 
<b>by </b><i><a class="ref" href="knaster.html#D3" target="_self">Def3</a>, , </i>;<br/>
<b>hence </b><a NAME="E19:54"/>
<font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div>
