<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2
 
<b>by </b><i><a class="ref" href="catalan1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:10"/><i><font color="Green">E11</font></i>: 
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="catalan1.html#E1:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="catalan1.html#E1:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="catalan1.html#E1:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="catalan1.html#E3:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1
<b>by </b><i><a class="txt" href="catalan1.html#E1:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<a NAME="E6:10"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="binom.html#K7">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</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">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan1.html#E2:10"><i><font color="Green">E10</font></i></a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/>
 <a href="catalan1.html#K1" target="_self">Catalan</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="binom.html#K7">choose</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalan1.html#E6:10"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="catalan1.html#E4:10"><i><font color="Green">E12</font></i></a>, <a class="ref" href="newton.html#T21">NEWTON:21</a></i>
;<br/>
<b>hence </b><a NAME="E8:10"/>
 <a href="catalan1.html#K1" target="_self">Catalan</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K6">!</a> </span>)</span></span>)</span>
 ;<br/>


</div>
