<?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>assume </b><a NAME="E1:49"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> ex <font color="Olive">C</font>, <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Olive">D</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> );<br/>
<a NAME="E2:49"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">C</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/>

<b>take </b><font color="Maroon">D</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/>

<b>thus </b> <a href="xboole_0.html#K1">{}</a>  = 
<a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>
;<br/>

<b>thus </b><a NAME="E2:49_1"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:49"/><i><font color="Green">E22</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="49_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>given </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:49_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2_1"/><i><font color="Green">E37</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>take </b><font color="Maroon">C1</font> = <font color="Maroon">C</font>;<br/><b>take </b><font color="Maroon">D1</font> =  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>;<br/><b>thus </b><a NAME="E2:49_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:49_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>;<br/><b>thus </b><a NAME="E3:49_2_1"/>
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E4:49"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2_2"/>
not  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:49_2_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/><a NAME="E3:49_2_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>take </b><font color="Maroon">C1</font> =  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font>;<br/><b>take </b><font color="Maroon">D1</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font> = 
<font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="txt" href="ordinal3.html#E4:49"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>
;<br/><b>thus </b><a NAME="E5:49_2_2"/>
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:49"/><i><font color="Green">E39</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">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">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</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="49_3"><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="E1:49_3"/><i><font color="Green">E40</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>  &amp; ( 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">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] ) )
 ;<br/>

<a NAME="E2:49_3"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, </i>;<br/>
<a NAME="E3:49_3"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<a NAME="E4:49_3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T59">ORDINAL2:59</a></i>;<br/>
<a NAME="E5:49_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> implies ( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, , <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_3_1"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:49_3_1"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E4:49"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:49_3_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:49_3_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">C</font>]
 <b>by </b><i>, <a class="ref" href="ordinal3.html#T19" target="_self">Th19</a></i>;<br/><b>consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:49_3_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">C</font>]
 <b>and </b><br/><a NAME="E7:49_3_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">C1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">C1</font>] holds <br/><font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C1</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="txt" href="ordinal3.html#E4:49_3"><i><font color="Green">E41</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_3_1_1"/>
<font color="Maroon">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/><b>then consider </b><font color="Maroon">C1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:49_3_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E5:49_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal3.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:49_3_1_1"/>
<font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E5:49_3"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><a NAME="E5:49_3_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:49_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">C1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:49_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C1</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><a NAME="E11:49_3_1"/>
<font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E12:49_3_1"/><b>then </b>
( <font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C1</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E5:49_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><a NAME="E13:49_3_1"/><b>then </b>
not <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E14:49_3_1"/><b>then </b>
<font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/><b>then consider </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E16:49_3_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E17:49_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">C1</font>
;<br/>

<b>take </b>
<font color="Maroon">D</font>
;<br/>

<b>thus </b><a NAME="E1:49_3_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E4:49_3_1"><i><font color="Green">E45</font></i></a></i>;<br/>

<a NAME="E2:49_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="ordinal3.html#E5:49_3"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="ordinal3.html#E4:49_3_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>;<br/>
<b>hence </b><a NAME="E3:49_3_1_2"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:49_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:49_3"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:49"/>
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="E6:49"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/>


</div>
