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

<b>let </b><font color="Maroon" title="c1">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>set </b><font color="Maroon" title="c2">p</font> = <font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0;<br/>
<a NAME="E1:35"/><i><font color="Green" title="E33">A1</font></i>: 
 <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,0 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,0
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">q</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c4">q</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="catalan2.html#E1:35_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></i>;<br/>
<a NAME="E4:35_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:35_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:35_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">q</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:35_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="stirl2_1.html#T53" title="STIRL2_1:th.53">STIRL2_1:53</a></i>;<br/>
<a NAME="E6:35_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:35_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="funcop_1.html#T17" title="FUNCOP_1:th.17">FUNCOP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E7:35_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="catalan2.html#E3:35_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_2"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:35_2"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 0 &amp; <font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0 is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green" title="E10">Lm2</font></i></a>, <a class="txt" href="catalan2.html#E11"><i><font color="Green" title="E11">Lm3</font></i></a></i>;<br/>
<a NAME="E3:35_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,0
 
<b>by </b><i><a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:35_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,0
 <b>by </b><i><a class="txt" href="catalan2.html#E1:35_2"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,0
 <b>by </b><i><a class="txt" href="catalan2.html#E1:35"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
