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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">a</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ) implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  )</span><br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">a</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ) )</span><br/><b>assume </b><a NAME="E1:4_1"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/><a NAME="E2:4_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> 
 <b>by </b><i><a class="txt" href="topgen_2.html#E1:4"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:4_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )
 <b>by </b><i><a class="txt" href="topgen_2.html#E1:4_1"><i><font color="Green" title="E6">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> is  <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> 
 <b>by </b><i><a class="ref" href="card_lar.html#T33" title="CARD_LAR:th.33">CARD_LAR:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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