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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">b</font>, <font color="Olive" title="b3">a</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">a</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="Olive" title="b1">f</font> holds <br/> for <font color="Olive" title="b4">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">O2</font>) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Olive" title="b3">a</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">a</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">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/> for <font color="Olive" title="b3">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Olive" title="b1">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">O2</font>) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">a</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">b</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> implies  for <font color="Olive" title="b1">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O2</font>) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c4">a</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O2</font>) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c4">a</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span><font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> (<font color="Maroon" title="c2">f</font>,$1) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c4">a</font>;<br/>
<a NAME="E3:47"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E41">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">O1</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">O1</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:47_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">O1</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">O1</font>]</span><br/><a NAME="E2:47_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">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> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font>) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">O1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="knaster.html#T16" target="_self" title="KNASTER:th.16">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E42">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">now </span></a><div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> = (<font color="Maroon" title="c2">f</font>,$1) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c4">a</font>;<br/><span class="kw">let </span><font color="Maroon" title="c5">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">O1</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b1">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">O1</font>] )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47_2"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">O1</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">and </span><br/><a NAME="E2:47_2"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
 for <font color="Olive" title="b1">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">O1</font>]</span><br/><span class="kw">consider </span><font color="Maroon" title="c6">L1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E4:47_2"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O1</font> &amp; (  for <font color="Olive" title="b1">O3</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O1</font> holds <br/><font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">O3</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></span><a NAME="E5:47_2"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">L1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">L1</font> or <font color="Maroon" title="c3">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_2_1"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">q</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">O3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:47_2_1"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
<font color="Maroon" title="c8">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">L1</font>
 <span class="kw">and </span><br/><a NAME="E4:47_2_1"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">O3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">O3</font> = <font color="Maroon" title="c8">O3</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_2_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a></span>;<br/>
<a NAME="E6:47_2_1"/>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">O3</font>) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E3:47_2_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:47_2_1"/>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E3:47_2_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E4:47_2_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:47_2"/>
(<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font>) <a href="knaster.html#K6" title="KNASTER:func.6">-.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">L1</font></span>)</span>,<font color="Maroon" title="c1">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="knaster.html#T18" target="_self" title="KNASTER:th.18">Th21</a></span>;<br/><span class="kw">hence </span><a NAME="E7:47_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">O1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_2"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="lattice3.html#T39" title="LATTICE3:th.39">LATTICE3:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:47"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="knaster.html#T14" target="_self" title="KNASTER:th.14">Th17</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:47"/>
 for <font color="Olive" title="b1">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E42">A5</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
