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<span class="kw">let </span><font color="Maroon" title="c1">e</font> be   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</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">  ex <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="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Olive" title="b1">a</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">U</font> =  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>;<br/>
<a NAME="E1:133"/><span class="lab"><font color="Green" title="E72">A0</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="ordinal6.html#T57" target="_self" title="ORDINAL6:th.57">Unc1</a></span>;<br/>
<a NAME="E2:133"/>
(  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>, <font color="Maroon" title="c4">g</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="ref" href="ordinal6.html#T62" target="_self" title="ORDINAL6:th.62">Th23</a></span>;<br/>
<span class="lab"><font color="Green" title="E73">A1</font></span>: <font color="Maroon" title="c4">g</font> = 
 <a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</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="ordinal6.html#T74" target="_self" title="ORDINAL6:th.74">Th12</a></span>
<br/>.= 
 <a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D15" target="_self" title="ORDINAL6:def.15">V</a></span>
;<br/>
<span class="lab"><font color="Green" title="E74">A2</font></span>: <font color="Maroon" title="c3">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c1">e</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">e</font>

<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<font color="Maroon" title="c1">e</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T68" target="_self" title="ORDINAL6:th.68">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c1">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#D5" title="ORDINAL5:def.5">ORDINAL5:def 5</a></span>
;<br/>
<a NAME="E6:133"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:133"/>
( <font color="Maroon" title="c1">e</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> &amp; <font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="classes1.html#T2" title="CLASSES1:th.2">CLASSES1:2</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E8:133"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A5</font></span>: 
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<a NAME="E9:133"/><span class="kw">then </span>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<a NAME="E10:133"/><span class="kw">then </span>
<font color="Maroon" title="c1">e</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E6:133"><span class="lab"><font color="Green" title="E75">A3</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="c5">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="E12:133"/><span class="lab"><font color="Green" title="E77">A4</font></span>: 
( <font color="Maroon" title="c5">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="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">e</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="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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/>
<span class="kw">take </span>
<font color="Maroon" title="c5">a</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="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">W</font> =  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>;<br/>
<a NAME="E13:133"/><span class="lab"><font color="Green" title="E78">A6</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> &amp; <font color="Maroon" title="c5">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="classes1.html#T2" title="CLASSES1:th.2">CLASSES1:2</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E14:133"/>
<font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>;<br/>
<a NAME="E15:133"/><span class="kw">then </span>
(  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp;  <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:133"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="txt" href="#E13:133"><span class="lab"><font color="Green" title="E78">A6</font></span></a>, <a class="ref" href="ordinal6.html#T57" target="_self" title="ORDINAL6:th.57">Unc1</a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:133"/>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="ref" href="ordinal6.html#T64" target="_self" title="ORDINAL6:th.64">Th25</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>
