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<span class="kw">let </span><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">N</font> be   <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"> ( not <font color="Maroon" title="c1">M</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; ( <font color="Maroon" title="c2">N</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font> or <font color="Maroon" title="c2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> ) implies ( <font color="Maroon" title="c1">M</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">N</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:144"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
not <font color="Maroon" title="c1">M</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">and </span><br/><a NAME="E2:144"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
( <font color="Maroon" title="c2">N</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font> or <font color="Maroon" title="c2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> )
 ; <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> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">N</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<a NAME="E3:144"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:144"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="card_3.html#T117" target="_self" title="CARD_3:th.117">Th117</a></span>;<br/>
<a NAME="E4:144"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c1">M</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"><font color="Maroon" title="c1">M</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#D1" title="CARD_2:def.1">CARD_2:def 1</a></span>;<br/>
<a NAME="E5:144"/><span class="lab"><font color="Green" title="E75">A5</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c2">N</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"><font color="Maroon" title="c1">M</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#D1" title="CARD_2:def.1">CARD_2:def 1</a></span>;<br/>
<a NAME="E6:144"/>
<font color="Maroon" title="c2">N</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="txt" href="#E2:144"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></span>;<br/>
<a NAME="E7:144"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A6</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">N</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T50" title="ORDINAL2:th.50">ORDINAL2:50</a></span>;<br/>
<a NAME="E8:144"/><span class="lab"><font color="Green" title="E77">A7</font></span>: 
<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> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></span>;<br/>
<a NAME="E9:144"/><span class="lab"><font color="Green" title="E78">A8</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:144"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E4:144"><span class="lab"><font color="Green" title="E74">A4</font></span></a></span>;<br/>
<a NAME="E10:144"/><span class="lab"><font color="Green" title="E79">A9</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c2">N</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="txt" href="#E3:144"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E4:144"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E5:144"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="txt" href="#E7:144"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<a NAME="E11:144"/>
<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> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:144"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="txt" href="#E8:144"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="txt" href="#E9:144"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:144"/>
( <font color="Maroon" title="c1">M</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">N</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:144"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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