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<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <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/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">a</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">b</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Olive" title="b1">f</font> holds <br/> ex <font color="Olive" title="b4">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b4">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="Olive" title="b1">f</font>,<font color="Olive" title="b4">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">b</font></span>)</span> &amp; <font color="Olive" title="b3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">b</font></span>)</span> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</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="Olive" title="b2">b</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> holds <br/> ex <font color="Olive" title="b3">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b3">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="Olive" title="b3">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</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="c4">b</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> implies  ex <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="Olive" title="b1">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <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="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> ) )</span><br/>







<b>assume </b><a NAME="E1:44"/>
( <font color="Maroon" title="c3">a</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="c4">b</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="Olive" title="b1">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <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="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> )</span><br/>

<a NAME="E2:44"/><b>then </b><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c3">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="c3">a</font> &amp; <font color="Maroon" title="c4">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="c4">b</font> )
 
<b>by </b><i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">ab</font> = <font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">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/>
<a NAME="E4:44"/><i><font color="Green" title="E36">A2</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">ab</font> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">ab</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/>
<a NAME="E5:44"/><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="c3">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="c5">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="c4">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="c5">ab</font> )
 
<b>by </b><i><a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</a></i>;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c5">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="c5">ab</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E2:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">O</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:44"/><i><font color="Green" title="E38">A4</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c5">ab</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="ref" href="knaster.html#T33" target="_self" title="KNASTER:th.33">Th33</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">O</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <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="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E9:44"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</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="txt" href="knaster.html#E8:44"><i><font color="Green" title="E38">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> )</span><br/>

<a NAME="E10:44"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c5">ab</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="knaster.html#E6:44"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="knaster.html#T25" target="_self" title="KNASTER:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E11:44"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="knaster.html#E4:44"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span></span><br/>

<b>thus </b><a NAME="E12:44"/>
<font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="knaster.html#E4:44"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="knaster.html#E10:44"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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