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<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="Red">::  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">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">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> holds <br/>for <font color="Olive" title="b3">O1</font>, <font color="Olive" title="b4">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b3">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b4">O2</font> holds <br/><font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <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">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Olive" title="b2">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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">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="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">O1</font>, <font color="Olive" title="b3">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b3">O2</font> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <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">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="Red">::  thesis: </font></span></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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> implies for <font color="Olive" title="b1">O1</font>, <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">O2</font> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">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>for <font color="Olive" title="b1">O1</font>, <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">O2</font> &amp; <font color="Olive" title="b2">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> $1 holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>;<br/>
<div><span class="lab"><font color="Green" title="E26">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">O4</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">O4</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="c4">O4</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &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="c4">O4</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O3</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O4</font>] )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
( <font color="Maroon" title="c4">O4</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="c4">O4</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">and </span><br/><a NAME="E2:38_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
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="c4">O4</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O3</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O4</font>]</span><br/><span class="kw">thus </span><a NAME="E3:38_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O4</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">O1</font>, <font color="Maroon" title="c6">O2</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O4</font> implies <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38_1_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c5">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">O2</font>
 <span class="kw">and </span><br/><a NAME="E2:38_1_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/>

<a NAME="E3:38_1_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
( <font color="Maroon" title="c6">O2</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">O4</font> iff ( <font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O4</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">O4</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E4:38_1_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
( <font color="Maroon" title="c5">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c6">O2</font> iff ( <font color="Maroon" title="c5">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c5">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">O2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_1_1"/>
( <font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O4</font> or <font color="Maroon" title="c6">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">O4</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:38_1_1_1_1"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_1_1_1_1_1"/>
( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> or <font color="Maroon" title="c5">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E4:38_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:38_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c5">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><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#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">L1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E3:38_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</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="c6">O2</font> holds <br/><font color="Maroon" title="c7">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="E4:38_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
( <font color="Maroon" title="c7">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c7">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">O1</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="c7">L1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E3:38_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:38_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">L1</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:38_1_1_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E3:38_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="knaster.html#T20" target="_self" title="KNASTER:th.20">Th20</a></span>;<br/><span class="kw">hence </span><a NAME="E6:38_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_2"/>
<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">O4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:38_1_1_1_2"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:38"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<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="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">O1</font>, <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">O2</font> holds <br/><font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Olive" title="b2">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/>

<div><span class="lab"><font color="Green" title="E28">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">O4</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O4</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="c4">O4</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:38_2"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O4</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c4">O4</font>]</span><br/><span class="kw">thus </span><a NAME="E2:38_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">O1</font>, <font color="Maroon" title="c6">O2</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> implies <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38_2_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c5">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">O2</font>
 <span class="kw">and </span><br/><a NAME="E2:38_2_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_2_1_1"/>
( ( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> ) or ( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> ) or ( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> ) or ( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_1"/>
( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:38_2_1_1_1"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_2"/>
( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><a NAME="E2:38_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">O2</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_2_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E3:38_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E4:38_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></span>;<br/><span class="kw">hence </span><a NAME="E5:38_2_1_1_2"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E1:38_2_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_3"/>
( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:38_2_1_1_3"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_4"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
( <font color="Maroon" title="c5">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><a NAME="E2:38_2_1_1_4"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_2_1_1_4_1_1"/>
( <font color="Maroon" title="c4">O4</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  ex <font color="Olive" title="b1">O3</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c4">O4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">O3</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_4_1_1_1"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<font color="Maroon" title="c4">O4</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:38_2_1_1_4_1_1_1"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_4_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_4_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_2_1_1_4_1_1_1_1_1"/>
( <font color="Maroon" title="c4">O4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">O4</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_4_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">O4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><a NAME="E2:38_2_1_1_4_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="knaster.html#T16" target="_self" title="KNASTER:th.16">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_2_1_1_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="knaster.html#T18" target="_self" title="KNASTER:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_4_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_4_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c4">O4</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><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#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">L1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E3:38_2_1_1_4_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O4</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="c4">O4</font> holds <br/><font color="Maroon" title="c7">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="E4:38_2_1_1_4_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">L1</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_4_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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="Red">:: according to </font></span><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">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="c7">L1</font> or <font color="Maroon" title="c8">q</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">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="c7">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">O3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:38_2_1_1_4_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c9">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="c7">L1</font>
 <span class="kw">and </span><br/><a NAME="E4:38_2_1_1_4_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">O3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">O3</font> = <font color="Maroon" title="c9">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="#E3:38_2_1_1_4_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a></span>;<br/>
<a NAME="E6:38_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c10">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E7:38_2_1_1_4_1_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c10">O3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E8:38_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c10">O3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_2_1_1_4_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E3:38_2_1_1_4_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E9:38_2_1_1_4_1_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">O3</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>;<br/>
<a NAME="E10:38_2_1_1_4_1_1_1_1_1_2_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">O3</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">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/>
<a NAME="E11:38_2_1_1_4_1_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">O3</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="knaster.html#T18" target="_self" title="KNASTER:th.18">Th18</a></span>;<br/>
<a NAME="E12:38_2_1_1_4_1_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A26</font></span>: 
<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">O3</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="knaster.html#T18" target="_self" title="KNASTER:th.18">Th18</a></span>;<br/>
<a NAME="E13:38_2_1_1_4_1_1_1_1_1_2_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">O3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_2_1_1_4_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E3:38_2_1_1_4_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/>
<a NAME="E14:38_2_1_1_4_1_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A27</font></span>: 
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">O3</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">O3</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E7:38_2_1_1_4_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a></span>;<br/>
<a NAME="E15:38_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">O3</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_2_1_1_4_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E3:38_2_1_1_4_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E4:38_2_1_1_4_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:38_2_1_1_4_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38_2_1_1_4_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E14:38_2_1_1_4_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:38_2_1_1_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">L1</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E1:38_2_1_1_4_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E3:38_2_1_1_4_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="knaster.html#T20" target="_self" title="KNASTER:th.20">Th20</a></span>;<br/><span class="kw">hence </span><a NAME="E6:38_2_1_1_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_2_1_1_4_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_4_1_1_2"/>
 ex <font color="Olive" title="b1">O3</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c4">O4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">O3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">O3</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:38_2_1_1_4_1_1_2"/><span class="lab"><font color="Green" title="E34">A28</font></span>: 
<font color="Maroon" title="c4">O4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">O3</font>
 ;<br/><a NAME="E4:38_2_1_1_4_1_1_2"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">O3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">O3</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">O3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></span>;<br/><a NAME="E5:38_2_1_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">O3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_2_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E6:38_2_1_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">O3</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>;<br/><a NAME="E7:38_2_1_1_4_1_1_2"/><span class="kw">then </span>
<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="c7">O3</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">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/><a NAME="E8:38_2_1_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_2_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="ref" href="knaster.html#T18" target="_self" title="KNASTER:th.18">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E9:38_2_1_1_4_1_1_2"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O4</font></span>)</span> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="knaster.html#T18" target="_self" title="KNASTER:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:38_2_1_1_4"/>
<font color="Maroon" title="c5">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c6">O2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E1:38_2_1_1_4"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E4:38_2_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E5:38_2_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2_1_1_4"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></span>;<br/><a NAME="E6:38_2_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c4">O4</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:38_2_1_1_4"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c6">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_2_1_1_4"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E2:38_2_1_1_4"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c4">O1</font>, <font color="Maroon" title="c5">O2</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">O2</font> implies <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font> )</span><br/>



<span class="kw">assume </span><a NAME="E4:38"/><span class="lab"><font color="Green" title="E29">A29</font></span>: 
<font color="Maroon" title="c4">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font></span><br/>

<a NAME="E5:38"/><span class="lab"><font color="Green" title="E30">A30</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E6:38"/>
for <font color="Olive" title="b1">O4</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">O4</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="#E5:38"><span class="lab"><font color="Green" title="E30">A30</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:38"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">O1</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</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>,<font color="Maroon" title="c5">O2</font> <a href="knaster.html#K5" title="KNASTER:func.5">+.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E29">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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