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

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">F</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E2:51_1"/><span class="lab"><font color="Green" title="E25">A0</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">F<sub>1</sub></font>() &amp; (  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 <font color="Olive" title="b1">a</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">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c1">F</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>1</sub></font>(<font color="Olive" title="b1">a</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></span>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_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="E1:51_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  or  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T4" title="ORDINAL6:th.4">ORDINAL6:4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">F<sub>1</sub></font>() is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E25">A0</font></span></a>, <a class="ref" href="finset_1.html#T10" title="FINSET_1:th.10">FINSET_1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="51_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_1_1_2"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">F<sub>1</sub></font>() is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>;<br/><a NAME="E2:51_1_1_2"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_2"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/><a NAME="E3:51_1_1_2"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 <a href="exchsort.html#K3" title="EXCHSORT:func.3">len-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T24" target="_self" title="EXCHSORT:th.24">TT0</a></span>;<br/><a NAME="E4:51_1_1_2"/><span class="lab"><font color="Green" title="E29">B1</font></span>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D11" title="ORDINAL1:def.11">ORDINAL1:def 11</a></span>;<br/><span class="lab"><font color="Green" title="E30">A4</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> = 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="txt" href="#E2:51_1_1_2"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>( <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> )
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E25">A0</font></span></a>, <a class="txt" href="#E4:51_1_1_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="txt" href="#E1:51_1_1_2"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>
;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>9</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ;<br/><a NAME="E6:51_1_1_2"/><span class="lab"><font color="Green" title="E31">J0</font></span>: 
<font color="Maroon">S<sub>9</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E23">C1</font></span></a>, <a class="txt" href="#E5:51_1_1_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>;<br/><a NAME="E7:51_1_1_2"/><span class="lab"><font color="Green" title="E32">J1</font></span>: 
 for <font color="Olive" title="b1">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>   st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:51_1_1_2_2"/><span class="lab"><font color="Green" title="E33">B4</font></span>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:51_1_1_2_2"/><span class="lab"><font color="Green" title="E34">B6</font></span>: 
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:51_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">B3</font></span>: 
( <font color="Maroon" title="c3">n</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="c1">F</font> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">n</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="c1">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</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="c3">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_2"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:51_1_1_2"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E4:51_1_1_2_2"/>
( <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) &amp; <font color="Maroon" title="c1">F</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="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">n</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E25">A0</font></span></a>, <a class="txt" href="#E1:51_1_1_2"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:51_1_1_2_2"><span class="lab"><font color="Green" title="E34">B6</font></span></a></span>;<br/>
<a NAME="E5:51_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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="c3">n</font></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E24">C2</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E25">A0</font></span></a>, <a class="txt" href="#E1:51_1_1_2_2"><span class="lab"><font color="Green" title="E33">B4</font></span></a>, <a class="txt" href="#E3:51_1_1_2_2"><span class="lab"><font color="Green" title="E35">B3</font></span></a></span>;<br/>
<a NAME="E6:51_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:51_1_1_2_2"/>
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_2_2"><span class="lab"><font color="Green" title="E33">B4</font></span></a>, <a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:51_1_1_2"/><span class="lab"><font color="Green" title="E33">JJ</font></span>: 
 for <font color="Olive" title="b1">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>  holds  <font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E6:51_1_1_2"><span class="lab"><font color="Green" title="E31">J0</font></span></a>, <a class="txt" href="#E7:51_1_1_2"><span class="lab"><font color="Green" title="E32">J1</font></span></a>);<br/></span><a NAME="E9:51_1_1_2"/>
 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 <font color="Olive" title="b1">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp;  <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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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="c3">a</font></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1_1_2_3"/><span class="lab"><font color="Green" title="E34">Z0</font></span>: 
( <font color="Maroon" title="c3">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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="c3">a</font></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c3">a</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1_2"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/>
<a NAME="E3:51_1_1_2_3"/><span class="lab"><font color="Green" title="E35">B4</font></span>: 
( <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c4">n</font>] &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</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="c3">a</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_2_3"><span class="lab"><font color="Green" title="E34">Z0</font></span></a>, <a class="txt" href="#E8:51_1_1_2"><span class="lab"><font color="Green" title="E33">JJ</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E4:51_1_1_2_3"/>
( <font color="Maroon" title="c1">F</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>1</sub></font>(<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon" title="c1">F</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="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">a</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E25">A0</font></span></a>, <a class="txt" href="#E1:51_1_1_2_3"><span class="lab"><font color="Green" title="E34">Z0</font></span></a>, <a class="txt" href="#E1:51_1_1_2"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:51_1_1_2"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:51_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <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="c3">a</font></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E24">C2</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E25">A0</font></span></a>, <a class="txt" href="#E3:51_1_1_2_3"><span class="lab"><font color="Green" title="E35">B4</font></span></a>, <a class="txt" href="#E1:51_1_1_2_3"><span class="lab"><font color="Green" title="E34">Z0</font></span></a>, <a class="txt" href="#E1:51_1_1_2"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:51_1_1_2"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:51_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> is  <a href="exchsort.html#V4" title="EXCHSORT:attr.4">descending</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_2"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="exchsort.html#T26" target="_self" title="EXCHSORT:th.26">DSC2</a>, <a class="txt" href="#E1:51_1_1_2"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/><a NAME="E11:51_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E23">C1</font></span></a>, <a class="txt" href="#E5:51_1_1_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="exchsort.html#T27" target="_self" title="EXCHSORT:th.27">TD1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:51_1_1_2"/>
<font color="Maroon">F<sub>1</sub></font>() is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1_2"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
