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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c1">A</font> implies 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c2">phi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E2:128"/><i><font color="Green" title="E67">A1</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; ( 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#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  holds <br/><font color="Maroon" title="c2">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i>
<i><font color="Green" title="E68">A2</font></i>: 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  = 
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">phi</font>
<b>by </b><i><a class="txt" href="#E2:128"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="card_3.html#E133"><i><font color="Green" title="E65">Lm2</font></i></a>, <a class="ref" href="ordinal2.html#T46" title="ORDINAL2:th.46">ORDINAL2:46</a></i>
<br/>.= 
 <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="c2">phi</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T35" title="ORDINAL2:th.35">ORDINAL2:35</a></i>
;<br/>
<a NAME="E4:128"/><i><font color="Green" title="E69">A3</font></i>: 
1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">B</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="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  or <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/>

<b>assume </b><a NAME="E1:128_2"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c4">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:128_2"/><i><font color="Green" title="E70">A4</font></i>: 
( <font color="Maroon" title="c4">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="c2">phi</font> &amp; <font color="Maroon" title="c3">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">C</font> )
 <b>by </b><i><a class="txt" href="#E3:128"><i><font color="Green" title="E68">A2</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="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:128_2"/><i><font color="Green" title="E71">A5</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">phi</font> &amp; <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="#E3:128_2"><i><font color="Green" title="E70">A4</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="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E5:128_2"><i><font color="Green" title="E71">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c6">x</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="#E2:128"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="#E5:128_2"><i><font color="Green" title="E71">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c6">x</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E2:128"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="#E5:128_2"><i><font color="Green" title="E71">A5</font></i></a></i>;<br/>
<a NAME="E9:128_2"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">x</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">x</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E10:128_2"/><b>then </b>
( <font color="Maroon" title="c8">x</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">x</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_2.html#T49" title="CARD_2:th.49">CARD_2:49</a></i>;<br/>
<a NAME="E11:128_2"/><b>then </b>
( <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">x</font> &amp; 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">x</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="#E2:128"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="#E5:128_2"><i><font color="Green" title="E71">A5</font></i></a>, <a class="txt" href="card_3.html#E133"><i><font color="Green" title="E65">Lm2</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<a NAME="E12:128_2"/><b>then </b>
<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> 
 
<b>by </b><i><a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></i>;<br/>
<b>hence </b><a NAME="E13:128_2"/>
<font color="Maroon" title="c3">B</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="#E3:128_2"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E5:128"/>
 <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="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E6:128"/><b>then </b><i><font color="Green" title="E70">A6</font></i>: 
 ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">B</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T30" title="ORDINAL3:th.30">ORDINAL3:30</a></i>;<br/>
<a NAME="E7:128"/>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></i>;<br/>
<a NAME="E8:128"/><b>then </b>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="#E4:128"><i><font color="Green" title="E69">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:128"/>
1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E6:128"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="ordinal3.html#T33" title="ORDINAL3:th.33">ORDINAL3:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
