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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E36">Th37</font></span>: <a NAME="T37"><span class="comment"><font color="firebrick">:: CATALAN2:37</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b3">CardF</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">pN</font> where <font color="Olive" title="b4">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  : ( <font color="Olive" title="b4">pN</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b5">N</font> where <font color="Olive" title="b5">N</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">pN</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Olive" title="b5">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">N</font> &amp; <font color="Olive" title="b5">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b3">CardF</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">CardF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> &amp; (  for <font color="Olive" title="b4">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">CardF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">j</font> <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="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">j</font></span>)</span>,<font color="Olive" title="b4">j</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</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> (<span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span> ) )</div></div>
