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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="abian.html#R1">is_a_fixpoint_of</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">P</font> = <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="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:49_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="knaster.html#E6:49_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E3:49_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:49_1_1"/>
<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="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">P</font> = <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="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E6:49_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">P</font> is <a href="knaster.html#V2" target="_self">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="knaster.html#D8" target="_self">KNASTER:def 8</a><br/>

<b>assume </b><a NAME="E1:49_1_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:49_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> )
 ;<br/>
<b>consider </b><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="E5:49_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <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">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">ab</font> = <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:49_1_1_1"/><i><font color="Green">E27</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> <font color="Maroon">ab</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">ab</font> )
 <b>by </b><i>, <a class="ref" href="knaster.html#T4" target="_self">Th4</a>, , </i>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">ab</font>;<br/>
<b>take </b>
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">ab</font>
;<br/>

<b>thus </b><a NAME="E10:49_1_1_1"/>
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">ab</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="knaster.html#D3" target="_self">Def3</a></i>;<br/>

<b>thus </b><a NAME="E11:49_1_1_1"/>
( <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">ab</font> )
 <b>by </b><i><a class="ref" href="knaster.html#T4" target="_self">Th4</a>, , <a class="ref" href="knaster.html#D3" target="_self">Def3</a></i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E12:49_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z'</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">zz</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:49_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">zz</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> )
 ;<br/>
<a NAME="E15:49_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="knaster.html#T4" target="_self">Th4</a>, , <a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/>
<a NAME="E16:49_1_1_1"/>
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ab</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E17:49_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ab</font> )
 
<b>by </b><i>, <a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/>
<a NAME="E18:49_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ab</font>
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<a NAME="E19:49_1_1_1"/>
<font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z'</font>
 
<b>by </b><i><a class="ref" href="knaster.html#T4" target="_self">Th4</a>, , , <a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<b>hence </b><a NAME="E20:49_1_1_1"/>
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z'</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:49_1_1"/>
<font color="Maroon">P</font> is <a href="knaster.html#V3" target="_self">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="knaster.html#D9" target="_self">KNASTER:def 9</a><br/>

<b>assume </b><a NAME="E1:49_1_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:49_1_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> )
 ;<br/>
<b>consider </b><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="E5:49_1_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <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">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">ab</font> = <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:49_1_1_2"/><i><font color="Green">E36</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#K6" target="_self">-.</a> <font color="Maroon">ab</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="knaster.html#T4" target="_self">Th4</a>, , <a class="txt" href="knaster.html#E9:49_1_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">ab</font>;<br/>
<b>take </b>
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">ab</font>
;<br/>

<b>thus </b><a NAME="E10:49_1_1_2"/>
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">ab</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="knaster.html#D3" target="_self">Def3</a></i>;<br/>

<b>thus </b><a NAME="E11:49_1_1_2"/>
( <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">ab</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="ref" href="knaster.html#D3" target="_self">Def3</a></i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E12:49_1_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">zz</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:49_1_1_2"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">zz</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> )
 ;<br/>
<a NAME="E15:49_1_1_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="knaster.html#T4" target="_self">Th4</a>, , <a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/>
<a NAME="E16:49_1_1_2"/>
( <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<a NAME="E17:49_1_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/>
<a NAME="E18:49_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ab</font>
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T7">FILTER_0:7</a></i>;<br/>
<a NAME="E19:49_1_1_2"/>
<font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ab</font>
 
<b>by </b><i><a class="ref" href="knaster.html#T4" target="_self">Th4</a>, , , <a class="ref" href="filter_0.html#T7">FILTER_0:7</a></i>;<br/>
<b>hence </b><a NAME="E20:49_1_1_2"/>
<font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K6" target="_self">-.</a> <font color="Maroon">ab</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as  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>by </b><i>, </i>;<br/>
<b>take </b>
 <a href="knaster.html#K7" target="_self">latt</a> <font color="Maroon">P</font>
;<br/>

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

<b>thus </b><a NAME="E9:49_1_1"/>
<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> 
 ;<br/>

<b>thus </b><a NAME="E10:49_1_1"/>
 <a href="knaster.html#K7" target="_self">latt</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="knaster.html#K7" target="_self">latt</a> <font color="Maroon">P</font>
 ;<br/>


</div>
