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<span class="kw">let </span><font color="Maroon" title="c1">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"> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">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">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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> $1;<br/>
<a NAME="E1:137"/><span class="lab"><font color="Green" title="E75">00</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ordinal6.html#E98"><span class="lab"><font color="Green" title="E72">Th16</font></span></a></span>;<br/>
<a NAME="E2:137"/><span class="lab"><font color="Green" title="E76">01</font></span>: 
 for <font color="Olive" title="b1">b</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">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">b</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ordinal6.html#E99"><span class="lab"><font color="Green" title="E73">Th17</font></span></a></span>;<br/>
<a NAME="E3:137"/><span class="lab"><font color="Green" title="E77">02</font></span>: 
 for <font color="Olive" title="b1">b</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">b</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">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b2">c</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="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">c</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ordinal6.html#E100"><span class="lab"><font color="Green" title="E74">Th18</font></span></a></span>;<br/>
<a NAME="E4:137"/>
 for <font color="Olive" title="b1">b</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="#E1:137"><span class="lab"><font color="Green" title="E75">00</font></span></a>, <a class="txt" href="#E2:137"><span class="lab"><font color="Green" title="E76">01</font></span></a>, <a class="txt" href="#E3:137"><span class="lab"><font color="Green" title="E77">02</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:137"/>
1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font>
 ; <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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