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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="ordinal4.html#M2" target="_self">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="ordinal2.html#V2">increasing</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="ordinal2.html#V3">continuous</a>
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green">E33</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E4:50"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2" target="_self">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E7:50"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>and </b><br/><a NAME="E8:50"/><i><font color="Green">E36</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E9:50"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E10:50"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S5">ORDINAL2:sch 5</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> );<br/>
<a NAME="E11:50"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E8:50"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E12:50"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:50_1"/><i><font color="Green">E41</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:50_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ordinal4.html#E1:50_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="ordinal4.html#E2:50_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E5:50_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E9:50"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal4.html#E2:50_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:50_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:50"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:50_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:50_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:50_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:50_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<a NAME="E5:50_2"/><b>then </b>
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:50_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E6:50_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E3:50_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:50"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal4.html#E11:50"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ordinal4.html#E12:50"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ordinal4.html#E13:50"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E15:50"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal4.html#E3:50_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ordinal4.html#E14:50"><i><font color="Green">E41</font></i></a>, <a class="txt" href="ordinal4.html#E3:50_3"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E6:50_3"/>
<font color="Maroon">c<sub>8</sub></font> <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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:50_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="ordinal4.html#E3:50_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E7:50_3"/>
<font color="Maroon">c<sub>5</sub></font> <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">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:50_3"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="ordinal2.html#NM1">Ordinal-Sequence</a> <b>by </b><i><a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:50_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ordinal4.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E2:50_4"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ordinal4.html#E4:50"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal4.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E3:50_4"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:50_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:50"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:50_5"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:50_5"/><b>then </b><i><font color="Green">E45</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T31">ORDINAL3:31</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E3:50_5"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E4:50_5"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_5_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:50_5_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:50_5_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>;<br/>
<a NAME="E4:50_5_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E1:50_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="ordinal4.html#E2:50_5_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> as    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E14:50"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E6:50_5_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E9:50"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal4.html#E17:50"><i><font color="Green">E42</font></i></a>, <a class="txt" href="ordinal4.html#E2:50_5_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="ordinal4.html#E3:50_5_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/>
<b>hence </b><a NAME="E7:50_5_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:50_5_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="ordinal4.html#E2:50_5_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="ordinal4.html#E3:50_5_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:50_5"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_5_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:50_5_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:50_5_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E4:50_5_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:50_5_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ordinal3.html#T29">ORDINAL3:29</a></i>;<br/>
<a NAME="E5:50_5_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:50_5_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a>, <a class="ref" href="ordinal3.html#T32">ORDINAL3:32</a></i>;<br/>
<a NAME="E6:50_5_2"/><b>then </b>
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:50_5_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:50_5_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:50_5"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal4.html#E3:50_5"><i><font color="Green">E46</font></i></a>, <a class="txt" href="ordinal4.html#E4:50_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="ordinal4.html#E5:50_5"><i><font color="Green">E48</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:50_5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E1:50_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="ordinal4.html#E2:50_5"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>
<a NAME="E19:50"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> &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">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E8:50"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<a NAME="E20:50"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E18:50"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal4.html#T16" target="_self">Th16</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E21:50"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_6"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_6"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal4.html#E3:50_6"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:50_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E14:50"><i><font color="Green">E41</font></i></a>, <a class="txt" href="ordinal4.html#E3:50_6"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:50_6"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:50_6"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ordinal4.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:50"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal4.html#T37" target="_self">Th37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E24:50"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E22:50"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ordinal4.html#D3" target="_self">Def3</a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E25:50"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:50"><i><font color="Green">E32</font></i></a>, <a class="txt" href="ordinal4.html#E20:50"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ordinal2.html#D17">ORDINAL2:def 17</a></i>;<br/>
<a NAME="E26:50"/>
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E24:50"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E27:50"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E28:50"/>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="ordinal2.html#V2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:50_7"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:50_7"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:50_7"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:50_7"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E4:50_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ordinal4.html#E1:50_7"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E50</font></i>:  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
 <a href="ordinal2.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ordinal4.html#E20:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="ordinal4.html#E27:50"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ordinal4.html#T8" target="_self">Th8</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal4.html#E25:50"><i><font color="Green">E48</font></i></a>, <a class="ref" href="ordinal2.html#D14">ORDINAL2:def 14</a></i>
;<br/>
<a NAME="E30:50"/>
 <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_9"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:50_9"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:50_9"><i><font color="Green">E51</font></i></a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:50_9"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:50_9"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal4.html#E6:50_9"><i><font color="Green">E53</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:50_9"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E27:50"><i><font color="Green">E49</font></i></a>, <a class="txt" href="ordinal4.html#E6:50_9"><i><font color="Green">E53</font></i></a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:50_9"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E9:50_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal4.html#E11:50_9"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E13:50_9"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E11:50_9"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<a NAME="E14:50_9"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E9:50"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal4.html#E6:50_9"><i><font color="Green">E53</font></i></a>, <a class="txt" href="ordinal4.html#E11:50_9"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ordinal4.html#E21:50"><i><font color="Green">E45</font></i></a>, <a class="txt" href="ordinal4.html#E9:50_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="ordinal4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E16:50_9"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>16</sub></font> iff ( <font color="Maroon">c<sub>12</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E17:50_9"/><b>then </b>
( ( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> ) or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E21:50"><i><font color="Green">E45</font></i></a>, <a class="txt" href="ordinal4.html#E24:50"><i><font color="Green">E47</font></i></a>, <a class="txt" href="ordinal4.html#E9:50_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E18:50_9"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a></i>;<br/>
<a NAME="E19:50_9"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E14:50_9"><i><font color="Green">E56</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E20:50_9"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E21:50_9"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:50_9"><i><font color="Green">E52</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:50"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K4" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E29:50"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<b>hence </b><a NAME="E32:50"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal4.html#E17:50"><i><font color="Green">E42</font></i></a></i>;<br/>


</div>
