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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> = <font color="Maroon">F<sub>3</sub></font>($1,<span class="p1">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>);<br/>
<b>consider </b><font color="Maroon" title="c1">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E2:44_1"/><i><font color="Green" title="E13">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E3:44_1"/><i><font color="Green" title="E14">A2</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> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon" title="c1">L</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>2</sub></font>() )
 <b>and </b><br/><a NAME="E4:44_1"/><i><font color="Green" title="E15">A3</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> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c1">L</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">L</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="E5:44_1"/><i><font color="Green" title="E16">A4</font></i>: 
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> <font color="Maroon">F<sub>1</sub></font>() &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> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> holds <br/><font color="Maroon" title="c1">L</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">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font></span>)</span>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S5" target="_self" title="ORDINAL2:sch.5">ORDINAL2:sch 5</a>();<br/></i>
<a NAME="E6:44_1"/>
<font color="Maroon" title="c1">L</font> is <a href="ordinal2.html#V1" title="ORDINAL2:attr.1">Ordinal-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c2">A</font> =  <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">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D8" target="_self" title="ORDINAL2:def.8">ORDINAL2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font></span><br/>

<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> <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">x</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">L</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c3">x</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">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:44_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
( <font color="Maroon" title="c4">y</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">L</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><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="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E3:44_1_1"><i><font color="Green" title="E18">A6</font></i></a></i>;<br/>
<div><i><font color="Green" title="E19">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c6">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E2:44_1_1_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a></span><br/><a NAME="E3:44_1_1_1"/>
<font color="Maroon" title="c1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">B</font>,<span class="p1">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="#E2:44_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E4:44_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E3:44_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="#E2:44_1_1_1"><i><font color="Green" title="E20">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1"/>
<font color="Maroon" title="c3">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>by </b><i><a class="txt" href="#E3:44_1_1"><i><font color="Green" title="E18">A6</font></i></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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_2"/><i><font color="Green" title="E20">A9</font></i>: 
( <font color="Maroon" title="c5">y</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; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a></span><br/><a NAME="E2:44_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">y</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/><a NAME="E3:44_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">y</font></span>)</span>)
 <b>by </b><i><a class="txt" href="#E2:44_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E5:44_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="#E3:44_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="#E1:44_1_1_2"><i><font color="Green" title="E20">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_2"/>
<font color="Maroon" title="c3">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>by </b><i><a class="txt" href="#E3:44_1_1"><i><font color="Green" title="E18">A6</font></i></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>
<a NAME="E7:44_1_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</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">L</font></span>)</span> &amp;  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</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">L</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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">L</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E2:44_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E3:44_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E1:44_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="#E3:44_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="#E5:44_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="ordinal2.html#D7" target="_self" title="ORDINAL2:def.7">Def7</a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E8:44_1_1"/>
<font color="Maroon" title="c3">x</font> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">L</font> = <font color="Maroon" title="c1">L</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> ;<br/>
<b>take </b><font color="Maroon" title="c3">fi</font> = <font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &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">F<sub>1</sub></font>() implies <font color="Maroon" title="c3">fi</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>2</sub></font>() ) &amp; ( 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> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c3">fi</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">F<sub>3</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span>) ) &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> <font color="Maroon">F<sub>1</sub></font>() &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> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> holds <br/><font color="Maroon" title="c3">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">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font></span>)</span>) ) )</span><br/>

<b>thus </b><a NAME="E8:44_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &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">F<sub>1</sub></font>() implies <font color="Maroon" title="c3">fi</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>2</sub></font>() ) )
 <b>by </b><i><a class="txt" href="#E2:44_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E3:44_1"><i><font color="Green" title="E14">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c3">fi</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">F<sub>3</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span>) ) &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> <font color="Maroon">F<sub>1</sub></font>() &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> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> holds <br/><font color="Maroon" title="c3">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">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font></span>)</span>) ) )</span><br/>

<b>thus </b><a NAME="E9:44_1"/>
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> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c3">fi</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">F<sub>3</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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> <font color="Maroon">F<sub>1</sub></font>() &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> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> holds <br/><font color="Maroon" title="c3">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">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font></span>)</span>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon" title="c3">fi</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="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font></span>)</span>) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c5">B</font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>
<br/>.= 
<font color="Maroon" title="c5">B</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T26" target="_self" title="ORDINAL2:th.26">Th26</a></i>
;<br/>
<b>hence </b><a NAME="E3:44_1_2"/>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon" title="c3">fi</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="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">A</font></span>)</span>) )
 <b>by </b><i><a class="txt" href="#E4:44_1"><i><font color="Green" title="E15">A3</font></i></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>thus </b><a NAME="E10:44_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> <font color="Maroon">F<sub>1</sub></font>() &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> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> holds <br/><font color="Maroon" title="c3">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">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font></span>)</span>)
 <b>by </b><i><a class="txt" href="#E5:44_1"><i><font color="Green" title="E16">A4</font></i></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>
