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<div class="add">

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E32">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds not <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> = <font color="Maroon">F<sub>1</sub></font>($2);<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<b>consider </b><font color="Maroon" title="c1">phi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>and </b><br/><a NAME="E4:40_1"/><i><font color="Green" title="E34">A5</font></i>: 
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  implies <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ) )
 <b>and </b><br/><a NAME="E5:40_1"/><i><font color="Green" title="E35">A6</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  holds <br/><font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font>)
 <b>and </b><br/><a NAME="E6:40_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  holds <br/><font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S11" title="ORDINAL2:sch.11">ORDINAL2:sch 11</a>();<br/></i>
<div><i><font color="Green" title="E36">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>now </b></a><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>not $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>($1);<br/><b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green" title="E37">A8</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c2">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:40_1_1"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 <b>and </b><br/><a NAME="E4:40_1_1"/><i><font color="Green" title="E39">A10</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="ordinal4.html#E1:40_1_1"><i><font color="Green" title="E37">A8</font></i></a>);<br/></i><a NAME="E5:40_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/><a NAME="E6:40_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>)) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>)
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:40"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/><a NAME="E7:40_1_1"/><b>then </b>
not <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>))
 <b>by </b><i><a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E8:40_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="ordinal4.html#E4:40_1_1"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E37">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><a NAME="E1:40_1_2"/>
( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>) &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>) )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:40_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="ordinal4.html#E7:40_1"><i><font color="Green" title="E36">A7</font></i></a></i>;<br/><a NAME="E2:40_1_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>)
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:40_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">A</font>)
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:40_1"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c1">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon" title="c3">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:40_1_3"/><i><font color="Green" title="E39">A13</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">phi</font> )
 ;<br/>

<a NAME="E2:40_1_3"/><b>then </b><i><font color="Green" title="E40">A14</font></i>: 
 ex <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T31" title="ORDINAL3:th.31">ORDINAL3:31</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1</span>)</span> );<br/>
<a NAME="E3:40_1_3"/><i><font color="Green" title="E41">A15</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E4:40_1_3"/><i><font color="Green" title="E42">A16</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">C</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_3_1"/><i><font color="Green" title="E43">A17</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">C</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:40_1_3_1"/><i><font color="Green" title="E44">A18</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<a NAME="E3:40_1_3_1"/><i><font color="Green" title="E45">A19</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">C</font></span>)</span> &amp; <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">C</font></span>)</span> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E5:40_1_3_1"/>
( <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c5">D</font>) &amp; <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c5">D</font>) &amp; <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E5:40_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="ordinal4.html#E8:40_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="ordinal4.html#E2:40_1_3_1"><i><font color="Green" title="E44">A18</font></i></a>, <a class="txt" href="ordinal4.html#E3:40_1_3_1"><i><font color="Green" title="E45">A19</font></i></a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></i>;<br/>
<b>hence </b><a NAME="E6:40_1_3_1"/>
<font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:40_1_3_1"><i><font color="Green" title="E43">A17</font></i></a>, <a class="txt" href="ordinal4.html#E2:40_1_3_1"><i><font color="Green" title="E44">A18</font></i></a>, <a class="txt" href="ordinal4.html#E3:40_1_3_1"><i><font color="Green" title="E45">A19</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:40_1_3"/><i><font color="Green" title="E43">A20</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_3_2"/><i><font color="Green" title="E44">A21</font></i>: 
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:40_1_3_2"/>
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">C</font></span>)</span>
 <b>and </b><br/><a NAME="E3:40_1_3_2"/><i><font color="Green" title="E45">A22</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<a NAME="E4:40_1_3_2"/>
<font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1_3_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="ref" href="ordinal3.html#T29" title="ORDINAL3:th.29">ORDINAL3:29</a></i>;<br/>
<a NAME="E5:40_1_3_2"/><b>then </b>
( <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:40_1_3_2"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a>, <a class="ref" href="ordinal3.html#T32" title="ORDINAL3:th.32">ORDINAL3:32</a></i>;<br/>
<a NAME="E6:40_1_3_2"/><b>then </b>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_3_2"/>
<font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1_3_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:40_1_3"/>
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal4.html#E3:40_1_3"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="ordinal4.html#E4:40_1_3"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="ordinal4.html#E5:40_1_3"><i><font color="Green" title="E43">A20</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:40_1_3"/>
<font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="ordinal4.html#E1:40_1_3"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="ordinal4.html#E2:40_1_3"><i><font color="Green" title="E40">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> = <font color="Maroon">F<sub>1</sub></font>($1);<br/>
<b>consider </b><font color="Maroon" title="c2">fi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E11:40_1"/><i><font color="Green" title="E39">A23</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font> holds <br/><font color="Maroon" title="c2">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">A</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E12:40_1"/>
( <font color="Maroon">H<sub>3</sub></font>( <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">phi</font> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">phi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="ordinal4.html#E4:40_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="ordinal4.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:40_1"/><b>then </b><i><font color="Green" title="E40">A24</font></i>: 
(  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="ordinal4.html#E9:40_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="ordinal4.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a>, <a class="ref" href="ordinal4.html#T16" target="_self" title="ORDINAL4:th.16">Th16</a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E14:40_1"/><b>then </b><i><font color="Green" title="E41">A25</font></i>: 
<font color="Maroon">H<sub>3</sub></font>( <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font>) <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c2">fi</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:40"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="ordinal4.html#E11:40_1"><i><font color="Green" title="E39">A23</font></i></a></i>;<br/>
<a NAME="E15:40_1"/>
<font color="Maroon" title="c2">fi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:40_1_4"/><i><font color="Green" title="E42">A26</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">fi</font> )
 ;<br/>

<a NAME="E2:40_1_4"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">fi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:40_1_4"/><b>then </b>
( <font color="Maroon" title="c2">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">A</font>) &amp; <font color="Maroon" title="c2">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c4">B</font>) )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E11:40_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="txt" href="ordinal4.html#E1:40_1_4"><i><font color="Green" title="E42">A26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:40_1_4"/>
<font color="Maroon" title="c2">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="ordinal4.html#E1:40_1_4"><i><font color="Green" title="E42">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E42">A27</font></i>:  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">fi</font> = 
 <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c2">fi</font>
<b>by </b><i><a class="txt" href="ordinal4.html#E11:40_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="txt" href="ordinal4.html#E13:40_1"><i><font color="Green" title="E40">A24</font></i></a>, <a class="ref" href="ordinal4.html#T8" target="_self" title="ORDINAL4:th.8">Th8</a></i>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>( <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font>)
<b>by </b><i><a class="txt" href="ordinal4.html#E14:40_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="ordinal2.html#D14" title="ORDINAL2:def.14">ORDINAL2:def 14</a></i>
;<br/>
<a NAME="E17:40_1"/>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">fi</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_6"/><i><font color="Green" title="E43">A28</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">fi</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:40_1_6"/><i><font color="Green" title="E44">A29</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">fi</font> &amp; <font color="Maroon" title="c4">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">B</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:40_1_6"><i><font color="Green" title="E43">A28</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:40_1_6"/><i><font color="Green" title="E45">A30</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">fi</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:40_1_6"><i><font color="Green" title="E44">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal4.html#E6:40_1_6"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:40_1_6"/><i><font color="Green" title="E46">A31</font></i>: 
( <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">phi</font> &amp; <font color="Maroon" title="c7">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E11:40_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="txt" href="ordinal4.html#E6:40_1_6"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:40_1_6"/><i><font color="Green" title="E47">A32</font></i>: 
( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">phi</font> &amp; <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E9:40_1_6"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c9">z</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal4.html#E11:40_1_6"><i><font color="Green" title="E47">A32</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E13:40_1_6"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="ordinal4.html#E11:40_1_6"><i><font color="Green" title="E47">A32</font></i></a>, <a class="txt" href="ordinal4.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<a NAME="E14:40_1_6"/><b>then </b><i><font color="Green" title="E48">A33</font></i>: 
( <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">C</font>) &amp; <font color="Maroon" title="c1">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">phi</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c7">y</font>) )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:40_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="ordinal4.html#E5:40_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="ordinal4.html#E11:40_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="txt" href="ordinal4.html#E6:40_1_6"><i><font color="Green" title="E45">A30</font></i></a>, <a class="txt" href="ordinal4.html#E11:40_1_6"><i><font color="Green" title="E47">A32</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:40_1_6"/>
( <font color="Maroon" title="c7">y</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c8">C</font> iff ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c7">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E16:40_1_6"/><b>then </b>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E9:40_1_6"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<a NAME="E17:40_1_6"/><b>then </b>
( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c7">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">C</font>) or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:40"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E18:40_1_6"/><b>then </b>
( <font color="Maroon" title="c5">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">C</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E14:40_1_6"><i><font color="Green" title="E48">A33</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E19:40_1_6"/><b>then </b>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E20:40_1_6"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:40_1_6"><i><font color="Green" title="E44">A29</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:40_1"/><b>then </b>
not  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>3</sub></font>( <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c1">phi</font>)
 
<b>by </b><i><a class="txt" href="ordinal4.html#E16:40_1"><i><font color="Green" title="E42">A27</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/>
<b>hence </b><a NAME="E19:40_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal4.html#E8:40_1"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/>


</div>
