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<span class="kw">let </span><font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c2">c</font>, <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="c1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> implies <font color="Maroon" title="c1">b</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <font color="Maroon" title="c3">a</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:127"/><span class="lab"><font color="Green" title="E67">Z0</font></span>: 
<font color="Maroon" title="c1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</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">b</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">U</font> =  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="E2:127"/><span class="lab"><font color="Green" title="E68">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="E3:127"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
( <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="c2">c</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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#T66" target="_self" title="ORDINAL6:th.66">LemT</a></span>;<br/>
<a NAME="E4:127"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A2</font></span>: 
<font color="Maroon" title="c1">b</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="txt" href="#E1:127"><span class="lab"><font color="Green" title="E67">Z0</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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> <font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c6">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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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> <font color="Maroon" title="c2">c</font> 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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="#E2:127"><span class="lab"><font color="Green" title="E68">A0</font></span></a>, <a class="ref" href="ordinal6.html#T66" target="_self" title="ORDINAL6:th.66">LemT</a>, <a class="ref" href="ordinal6.html#T62" target="_self" title="ORDINAL6:th.62">Th23</a></span>;<br/>
<a NAME="E6:127"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="E7:127"/><span class="kw">then </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> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<a NAME="E8:127"/><span class="kw">then </span>
<font color="Maroon" title="c6">g</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c3">a</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E67">Z0</font></span></a>, <a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E68">A0</font></span></a>, <a class="ref" href="ordinal6.html#T66" target="_self" title="ORDINAL6:th.66">LemT</a>, <a class="ref" href="ordinal6.html#T58" target="_self" title="ORDINAL6:th.58">Th51</a></span>;<br/>
<a NAME="E9:127"/><span class="kw">then </span>
<font color="Maroon" title="c6">g</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c3">a</font></span>)</span>
 
<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/>
<span class="kw">hence </span><a NAME="E10:127"/>
<font color="Maroon" title="c1">b</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="ordinal6.html#K12" title="ORDINAL6:func.12">-Veblen</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E68">A0</font></span></a>, <a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="ordinal6.html#T67" target="_self" title="ORDINAL6:th.67">Th26</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/>


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