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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">M</font> is  <a href="card_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
not <font color="Maroon" title="c1">M</font> is  <a href="card_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:59_1_1"/>
not <font color="Maroon" title="c1">M</font> is  <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#D4" title="CARD_5:def.4">CARD_5:def 4</a></span>;<br/>
<a NAME="E4:59_1_1"/>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#D2" title="CARD_5:def.2">CARD_5:def 2</a></span>;<br/>
<a NAME="E5:59_1_1"/><span class="kw">then </span>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:59_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="card_5.html#T40" title="CARD_5:th.40">CARD_5:40</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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