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

<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 implies  ex <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font> );<br/>
<a NAME="E1:97"/><span class="lab"><font color="Green" title="E67">A0</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st ( for <font color="Olive" title="b2">b</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">b</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:97_1"/><span class="lab"><font color="Green" title="E68">Z0</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 <span class="kw">and </span><br/><a NAME="E2:97_1"/><span class="lab"><font color="Green" title="E69">Z1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> , <font color="Maroon" title="c4">b</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:97_1"/><span class="lab"><font color="Green" title="E70">Z2</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p3">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_1"><span class="lab"><font color="Green" title="E69">Z1</font></span></a>, <a class="ref" href="ordinal5.html#T62" target="_self" title="ORDINAL5:th.62">Th43</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p2">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> as   <a href="ordinal5.html#V5" title="ORDINAL5:attr.5">Cantor-component</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E70">Z2</font></span></a>, <a class="ref" href="ordinal5.html#D9" target="_self" title="ORDINAL5:def.9">CC</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">c</font> = <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E6:97_1"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_1"><span class="lab"><font color="Green" title="E69">Z1</font></span></a>, <a class="ref" href="ordinal5.html#D10" target="_self" title="ORDINAL5:def.10">EXP</a></span>;<br/>
<a NAME="E7:97_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">Z3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">b</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_1"><span class="lab"><font color="Green" title="E68">Z0</font></span></a>, <a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E70">Z2</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:97_1_1"/>
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:97_1_1_1"/><span class="lab"><font color="Green" title="E72">B1</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c7">A</font> = <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p4">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E70">Z2</font></span></a>, <a class="ref" href="ordinal5.html#T65" target="_self" title="ORDINAL5:th.65">Th44</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Maroon" title="c7">A</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c2">a</font> = 
<font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p2">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E70">Z2</font></span></a>, <a class="txt" href="#E1:97_1_1_1"><span class="lab"><font color="Green" title="E72">B1</font></span></a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></span>
<br/>.= 
 <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Maroon" title="c7">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T53" target="_self" title="ORDINAL5:th.53">Th51</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:97_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">A</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E3:97_1_1_2"/><span class="lab"><font color="Green" title="E72">B3</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:97_1"><span class="lab"><font color="Green" title="E71">Z3</font></span></a></span>;<br/><a NAME="E4:97_1_1_2"/><span class="lab"><font color="Green" title="E73">B4</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E70">Z2</font></span></a>, <a class="txt" href="#E3:97_1_1_2"><span class="lab"><font color="Green" title="E72">B3</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="ordinal5.html#T56" target="_self" title="ORDINAL5:th.56">Th53</a></span>;<br/><a NAME="E5:97_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/><a NAME="E6:97_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="ordinal5.html#V5" title="ORDINAL5:attr.5">Cantor-component</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#D11" target="_self" title="ORDINAL5:def.11">CNF</a></span>;<br/><a NAME="E7:97_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/><a NAME="E8:97_1_1_2"/><span class="kw">then </span>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#D10" target="_self" title="ORDINAL5:def.10">EXP</a></span>;<br/><a NAME="E9:97_1_1_2"/><span class="kw">then </span>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1_1_2"><span class="lab"><font color="Green" title="E73">B4</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/><a NAME="E10:97_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">B5</font></span>: 
<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T12" target="_self" title="ORDINAL5:th.12">Th006</a></span>;<br/><a NAME="E11:97_1_1_2"/>
<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E70">Z2</font></span></a>, <a class="ref" href="ordinal5.html#T58" target="_self" title="ORDINAL5:th.58">Th42</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> = <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">s</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">A</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:97_1_1_2"><span class="lab"><font color="Green" title="E74">B5</font></span></a>, <a class="ref" href="ordinal5.html#T68" target="_self" title="ORDINAL5:th.68">Th45</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Maroon" title="c8">B</font></span><br/><span class="kw">thus </span><a NAME="E13:97_1_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E70">Z2</font></span></a>, <a class="txt" href="#E3:97_1_1_2"><span class="lab"><font color="Green" title="E72">B3</font></span></a>, <a class="ref" href="ordinal5.html#T55" target="_self" title="ORDINAL5:th.55">Th52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:97"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E67">A0</font></span></a>);<br/></span>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:97_2"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_2_1"><span class="kw">suppose </span></a><a NAME="E1:97_2_1"/><span class="lab"><font color="Green" title="E69">Y0</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c2">A</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c2">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Maroon" title="c2">A</font></span><br/><span class="kw">thus </span><a NAME="E3:97_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_2_1"><span class="lab"><font color="Green" title="E69">Y0</font></span></a>, <a class="ref" href="ordinal5.html#T52" target="_self" title="ORDINAL5:th.52">Th50</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_2_2"><span class="kw">suppose </span></a><a NAME="E1:97_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font></span><br/><div class="add"><a NAME="E2:97_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E68">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:97_2_2"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal5.html#V6" title="ORDINAL5:attr.6">Cantor-normal-form</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K4" title="ORDINAL5:func.4">Sum^</a> <font color="Olive" title="b1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
