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

<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>let </b><font color="Maroon">p</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">q</font> = <font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1;<br/>
<b>assume </b><a NAME="E1:19"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:19"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="zfmisc_1.html#T12">ZFMISC_1:12</a></i>;<br/>
<a NAME="E3:19"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, </i>;<br/>
<a NAME="E4:19"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:19"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T29">AFINSQ_1:29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="catalan2.html#D1" target="_self">CATALAN2:def 1</a><br/>

<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>assume </b><a NAME="E6:19"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:19_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:19_1_1_1"/>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">md</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span> 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/><a NAME="E3:19_1_1_2"/>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:19_1_1_2"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E5:19_1_1_2"/><b>then </b>
<font color="Maroon">md</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:19_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">md</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#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:19_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">md</font> <a href="hidden.html#R1">=</a> <font color="Maroon">md</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1 &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">md</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">md</font> <a href="binop_2.html#K24">*</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">md</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:19_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">md</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">md</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">md</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E9:19_1_1_2"/><b>then </b>
( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font></span>)</span></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="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">md</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:19_1_1_2"/><b>then </b>
( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> 0 &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><b>hence </b><a NAME="E11:19_1_1_2"/>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:19"/>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="afinsq_1.html#K5">^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>


</div>
