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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green">E51</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:41_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="newton.html#T29">NEWTON:29</a></i>;<br/><a NAME="E3:41_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><b>hence </b><a NAME="E4:41_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><a NAME="E3:41_1_1_2"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">m1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:41_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">nm</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/><b>set </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> <a href="newton.html#K12">!</a> ;<br/><b>set </b><font color="Maroon">m'</font> = <font color="Maroon">m</font> <a href="newton.html#K12">!</a> ;<br/><b>set </b><font color="Maroon">nm'</font> = <font color="Maroon">nm</font> <a href="newton.html#K12">!</a> ;<br/><b>set </b><font color="Maroon">nm1'</font> = <span class="p1">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> ;<br/><b>set </b><font color="Maroon">m1'</font> = <font color="Maroon">m1</font> <a href="newton.html#K12">!</a> ;<br/><i><font color="Green">E57</font></i>: 1 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
;<br/><i><font color="Green">E58</font></i>: 1 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">m</font> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T191">XCMPLX_1:191</a></i>
;<br/><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>
;<br/><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/><a NAME="E10:41_1_1_2"/>
<font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E11:41_1_1_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/><b>hence </b><a NAME="E12:41_1_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:41"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>


</div>
