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<div><i><font color="Green" title="E21">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c2">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E2:31_1_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c1">A</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="c2">B</font>
 ;<br/><a NAME="E3:31_1_1_1"/>
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</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"><span class="p3">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c2">B</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="card_1.html#E2:31_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="card_1.html#E46"><i><font color="Green" title="E20">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:31_1_1_1"/>
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c1">A</font> is <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_1_2"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c1">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; ( 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="c1">A</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> ) )
 ;<br/><a NAME="E2:31_1_1_2"/><b>then </b><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/><b>consider </b><font color="Maroon" title="c2">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E4:31_1_1_2"/><i><font color="Green" title="E24">A5</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i><a NAME="E5:31_1_1_2"/>
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_1.html#E1:31_1_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="card_1.html#E2:31_1_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="card_1.html#E4:31_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="card_1.html#E46"><i><font color="Green" title="E20">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E6:31_1_1_2"/>
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c1">A</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_1_1"/>
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c1">A</font> is <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>
 <b>by </b><i><a class="txt" href="card_1.html#E1:31_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="card_1.html#E46"><i><font color="Green" title="E20">Lm2</font></i></a></i>;<br/>


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