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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">O</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="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O</font></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span><font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,$1;<br/>
<a NAME="E1:29"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
for <font color="Olive" title="b1">O1</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">O1</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">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">O1</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">O1</font>] )</span><br/>

<a NAME="E1:29_1"/>
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="c3">O1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">O1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#T16" target="_self" title="LATTICE8:th.16">Th16</a></span>;<br/>
<a NAME="E2:29_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="c3">O1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:29_1"/>
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">O1</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="c3">O1</font>]</span><br/>

<span class="kw">hence </span><a NAME="E4:29_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">O1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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="E2:29"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">O1</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">O2</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,$1;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="c3">O2</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">O2</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29_2"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c3">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> 
 <span class="kw">and </span><br/><a NAME="E2:29_2"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<font color="Maroon" title="c3">O2</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">and </span><br/><a NAME="E3:29_2"/>
for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O2</font> holds <br/><font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O1</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="c3">O2</font>]</span><br/>

<a NAME="E4:29_2"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">Ls</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E6:29_2"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">Ls</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">O2</font> &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O2</font> holds <br/><font color="Maroon" title="c4">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">O1</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>
<font color="Maroon" title="c4">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#T15" target="_self" title="LATTICE8:th.15">Th15</a></span>
;<br/>
<a NAME="E8:29_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c1">A</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="c4">Ls</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E4:29_2"><span class="lab"><font color="Green" title="E26">A6</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="E9:29_2"/>
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">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="c4">Ls</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="lattice8.html#T17" target="_self" title="LATTICE8:th.17">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:29_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">O2</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_2"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</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="E3:29"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#T15" target="_self" title="LATTICE8:th.15">Th15</a></span>;<br/>
<a NAME="E4:29"/>
for <font color="Olive" title="b1">O</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">O</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="#E3:29"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E23">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:29"/>
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">O</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>
