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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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/> <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b2">O2</font></span><br/><b>let </b><font color="Maroon" title="c2">O1</font>, <font color="Maroon" title="c3">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">O2</font> implies  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">O2</font> )</span><br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( <font color="Maroon" title="c2">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> $1 implies  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,$1 );<br/>
<a NAME="E1:59"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E2:59"/><i><font color="Green" title="E45">A2</font></i>: 
for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">O2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O2</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">O2</font>] )</span><br/>

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green" title="E46">A3</font></i>: 
( <font color="Maroon" title="c2">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O2</font> implies  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">O2</font>]</span><br/>

<b>assume </b><a NAME="E2:59_1"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c2">O1</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">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O2</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1"/>
( <font color="Maroon" title="c2">O1</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">O2</font> or <font color="Maroon" title="c2">O1</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">O2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1"/>
<font color="Maroon" title="c2">O1</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">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O2</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:59_1_1_1"/>
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_2"/>
<font color="Maroon" title="c2">O1</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">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O2</font></span>)</span></span><br/><div class="add"><a NAME="E2:59_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">O1</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">O2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:59_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:59_1_1_2"/><b>then </b><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c2">O1</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">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E4:59_1_1_2"/>
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:59_1_1_2"/><b>then </b>
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:59_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="lattice5.html#E3:59_1_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:59_1_1_2"/>
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">O2</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice5.html#T25" target="_self" title="LATTICE5:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:59"/><i><font color="Green" title="E46">A6</font></i>: 
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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">O2</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">O3</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">O3</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">O2</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">O2</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_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">O2</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">O2</font>] )</span><br/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green" title="E47">A7</font></i>: 
( <font color="Maroon" title="c4">O2</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">O2</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_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">O2</font> &amp; <font color="Maroon" title="c2">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">O3</font> holds <br/> <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O3</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">O2</font>]</span><br/>

<b>assume </b><a NAME="E2:59_2"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c2">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font></span><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c5">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E4:59_2"/><i><font color="Green" title="E49">A9</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c4">O2</font> holds <br/><font color="Maroon" title="c5">L</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>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E5:59_2"/><i><font color="Green" title="E50">A10</font></i>: 
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:59_2"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="lattice5.html#E4:59_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="lattice5.html#T26" target="_self" title="LATTICE5:th.26">Th26</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1"/>
( <font color="Maroon" title="c2">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> or <font color="Maroon" title="c2">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">O2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1"><b>suppose </b></a><a NAME="E1:59_2_1_1"/>
<font color="Maroon" title="c2">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font></span><br/><div class="add"><b>hence </b><a NAME="E2:59_2_1_1"/>
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2"><b>suppose </b></a><a NAME="E1:59_2_1_2"/>
<font color="Maroon" title="c2">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font></span><br/><div class="add"><a NAME="E2:59_2_1_2"/><b>then </b><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c2">O1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">O2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:59_2"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:59_2_1_2"/><b>then </b><i><font color="Green" title="E52">A12</font></i>: 
<font color="Maroon" title="c2">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E4:59_2_1_2"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:59_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="lattice5.html#E2:59_2_1_2"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E5:59_2_1_2"/>
<font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">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="c5">L</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:59_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="lattice5.html#E3:59_2_1_2"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:59_2_1_2"/>
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">O2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:59_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="lattice5.html#E4:59_2_1_2"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:59"/>
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>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="lattice5.html#E1:59"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="lattice5.html#E2:59"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="lattice5.html#E3:59"><i><font color="Green" title="E46">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:59"/>
( <font color="Maroon" title="c2">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">O2</font> implies  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">O2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
