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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<a NAME="E1:18"/><i><font color="Green">E15</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">H<sub>3</sub></font>( <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>, <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>) <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>) <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="card_2.html#E8"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<a NAME="E2:18"/>
( <font color="Maroon">c<sub>1</sub></font>, <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">c<sub>2</sub></font>, <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E3:18"/><b>then </b>
( <font color="Maroon">H<sub>3</sub></font>( <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>, <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>),<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>),<font color="Maroon">H<sub>3</sub></font>( <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>, <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>) <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="card_2.html#E8"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="card_2.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E4:18"/><b>then </b>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>),<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<a NAME="E5:18"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="card_2.html#K1" target="_self">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="card_2.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<b>hence </b><a NAME="E6:18"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="card_2.html#K1" target="_self">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


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