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

<span class="kw">let </span><font color="Maroon" title="c1">l</font> be   <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st <font color="Olive" title="b1">f</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  &amp; <font color="Maroon" title="c1">l</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"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Olive" title="b1">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</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="c2">f</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  &amp; <font color="Maroon" title="c1">l</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"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> implies <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">g</font> =  <a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">h</font> = <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font> as   <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">X</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font>;<br/>
<span class="kw">assume </span><a NAME="E2:62"/><span class="lab"><font color="Green" title="E30">A0</font></span>: 
( <font color="Maroon" title="c2">f</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  &amp; <font color="Maroon" title="c1">l</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"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></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"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span></span><br/>

<a NAME="E3:62"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T29" target="_self" title="ORDINAL6:th.29">Th02</a></span>;<br/>
<a NAME="E4:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</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">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/>
<a NAME="E5:62"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c1">l</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E30">A0</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E6:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E7:62"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c4">h</font> holds <br/><font color="Olive" title="b1">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c6">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="c4">h</font> implies <font color="Maroon" title="c6">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_1"/>
<font color="Maroon" title="c6">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="c4">h</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" title="c6">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:62_1"/><span class="lab"><font color="Green" title="E35">B1</font></span>: 
( <font color="Maroon" title="c7">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="c4">h</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:62_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E3:62_1"><span class="lab"><font color="Green" title="E35">B1</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:62_1"/>
<font color="Maroon" title="c6">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T29" target="_self" title="ORDINAL6:th.29">Th02</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>
<span class="kw">reconsider </span><font color="Maroon" title="c6">u</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font></span>)</span> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E9:62"/><span class="lab"><font color="Green" title="E35">S5</font></span>: 
<font color="Maroon" title="c4">h</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c7">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="c4">h</font> implies <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:62_2"/>
<font color="Maroon" title="c7">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="c4">h</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" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:62_2"/><span class="lab"><font color="Green" title="E36">B2</font></span>: 
( <font color="Maroon" title="c8">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="c4">h</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:62_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E3:62_2"><span class="lab"><font color="Green" title="E36">B2</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E5:62_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E30">A0</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E3:62_2"><span class="lab"><font color="Green" title="E36">B2</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:62_2"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</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="E11:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A9</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T76" title="ZFMISC_1:th.76">ZFMISC_1:76</a></span>;<br/>
<a NAME="E12:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c6">u</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">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<a NAME="E13:62"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E30">A0</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E35">S5</font></span></a>, <a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="ordinal6.html#T37" target="_self" title="ORDINAL6:th.37">Th08</a>, <a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<a NAME="E14:62"/><span class="kw">then </span>
<font color="Maroon" title="c6">u</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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> <span class="kw"> such that </span><br/><a NAME="E16:62"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
( <font color="Maroon" title="c7">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"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T33" target="_self" title="ORDINAL6:th.33">Th06</a></span>;<br/>
<a NAME="E17:62"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:62_3"/>
<font color="Maroon" title="c7">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E30">A0</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="ordinal6.html#T22" target="_self" title="ORDINAL6:th.22">ThN4</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="c1">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:62_3"/><span class="lab"><font color="Green" title="E39">C1</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</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" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font></span><br/>

<a NAME="E3:62_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">C2</font></span>: 
 <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> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T28" title="ORDINAL1:th.28">ORDINAL1:28</a></span>;<br/>
<a NAME="E4:62_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">C3</font></span>: 
( <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> &amp; <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></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="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</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="c8">x</font></span>)</span> &amp; <font color="Maroon" title="c4">h</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="c8">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:62_3"><span class="lab"><font color="Green" title="E39">C1</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:62_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a></span>;<br/>
<a NAME="E6:62_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">h</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="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E3:62_3"><span class="lab"><font color="Green" title="E40">C2</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/>
<a NAME="E7:62_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_3"><span class="lab"><font color="Green" title="E41">C3</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E8:62_3"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E2:62_3"><span class="lab"><font color="Green" title="E39">C1</font></span></a>, <a class="ref" href="ordinal6.html#T4" target="_self" title="ORDINAL6:th.4">EXCH1</a></span>;<br/>
<a NAME="E9:62_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="ordinal6.html#T22" target="_self" title="ORDINAL6:th.22">ThN4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:62_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T4" target="_self" title="ORDINAL6:th.4">EXCH1</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>
<span class="kw">hence </span><a NAME="E18:62"/>
<span class="p1">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E38">A8</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>
