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

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">B</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>for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="card_1.html#K3" target="_self">alef</a> <font color="Olive">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:33"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E2:33"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/>

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

<b>assume </b><a NAME="E2:33_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E3:33_1"/><i><font color="Green">E49</font></i>: 
 <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2" target="_self">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:33_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">B</font> iff ( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:33_1_1"/><b>then </b>
(  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> &amp;  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K2" target="_self">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:33_1_1"/>
 <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:33_1"/>
 <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:33"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:33_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:33_2"/>
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/>for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> holds <br/> <a href="card_1.html#K3" target="_self">alef</a> <font color="Olive">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Olive">C</font>
 ;<br/>

<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E4:33_2"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:33_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">C</font>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E6:33_2"/><i><font color="Green">E58</font></i>: 
 <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1" target="_self">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E7:33_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E8:33_2"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<a NAME="E9:33_2"/><b>then </b>
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:33_2"/><b>then </b>
(  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a>, <a class="ref" href="ordinal2.html#T35">ORDINAL2:35</a></i>;<br/>
<a NAME="E11:33_2"/><b>then </b>
(  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font> &amp;  <a href="card_1.html#K1" target="_self">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" target="_self">alef</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E12:33_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:33_2"/>
(  <a href="card_1.html#K3" target="_self">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2" target="_self">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K2" target="_self">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:33_2"/>
 <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:33"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, , );<br/></i>
<b>hence </b><a NAME="E5:33"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> )
 ;<br/>

<b>assume </b><a NAME="E6:33"/><i><font color="Green">E62</font></i>: 
 <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E7:33"/><b>then </b><i><font color="Green">E63</font></i>: 
not <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:33"/>
 <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:33"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>


</div>
