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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">O</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b><font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:50"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i/>;<br/>
<a NAME="E2:50"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font>
 ;<br/>

<a NAME="E2:50_1"/>
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <span class="p1">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:50_1"/><b>then </b>
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:50_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:50"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">O2</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Olive">O2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O1</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">O2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">O2</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O1</font> ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">Ls</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:50_2"/><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font> &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">O1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E4:50_2"/><i><font color="Green">E50</font></i>: 
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:50_2"/><i><font color="Green">E51</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font>
 
<b>by </b><i>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<b>then </b><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  = 
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<a href="xboole_0.html#K1">{}</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i/>
;<br/>
<a NAME="E7:50_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 
<b>by </b><i>, <a class="txt" href="lattice5.html#E3:50"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:50_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:50"/>
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, <a class="txt" href="lattice5.html#E2:50"><i><font color="Green">E37</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E5:50"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O</font>
 ;<br/>


</div>
