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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E38">A1</font></i>: 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">CH</font> =  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,0;<br/>
<b>set </b><font color="Maroon" title="c4">Z</font> =  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">W</font> =  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">m</font>,1,0;<br/>
<a NAME="E2:40"/>
 <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,0
 
<b>by </b><i><a class="ref" href="catalan2.html#T25" target="_self" title="CATALAN2:th.25">Th25</a></i>;<br/>
<a NAME="E3:40"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,0</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/>
<a NAME="E4:40"/><b>then </b>
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,0</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c2">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> &amp; 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
;<br/>
<a NAME="E5:40"/><b>then </b>
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,0</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">m</font>,1,0</span>)</span></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">m</font>,1,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:40"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="catalan2.html#T31" target="_self" title="CATALAN2:th.31">Th31</a>, <a class="ref" href="card_fin.html#T18" title="CARD_FIN:th.18">CARD_FIN:18</a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>
<b>hence </b><a NAME="E6:40"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c1">m</font></span>)</span>
 ;<br/>


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