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<div><span class="lab"><font color="Green" title="E20">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c2">S</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E2:48_1_1_1"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">S</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">S</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>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></span><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:48_1_1_1"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_1"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><br/><a NAME="E5:48_1_1_1"/>
<font color="Maroon" title="c1">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></span>;<br/><a NAME="E6:48_1_1_1"/><span class="kw">then </span>
 <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">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E2:48_1_1_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="card_1.html#E45"><span class="lab"><font color="Green" title="E19">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_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#NM1" title="CARD_1:NM.1">Cardinal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c2">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><a NAME="E2:48_1_1_2"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/><a NAME="E3:48_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#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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="card_1.html#E45"><span class="lab"><font color="Green" title="E19">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_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#V1" title="CARD_1:attr.1">cardinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:48_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="card_1.html#E45"><span class="lab"><font color="Green" title="E19">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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