<?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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>thus </b><a NAME="E1:34"/>
(  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> is <a href="xboole_0.html#V1">empty</a> implies 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E51</font></i>: 
 <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>assume </b><a NAME="E2:34_1"/><i><font color="Green">E53</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/>

<a NAME="E3:34_1"/>
<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>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E4:34_1"/><b>then </b>
<font color="Maroon">m</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">p</font> = <font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">q</font> = <font color="Maroon">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1;<br/>
<a NAME="E6:34_1"/>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">m</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> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</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">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">nm</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K24">*</a> <font color="Maroon">nm</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">m</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> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nm</font> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, , , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E7:34_1"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">nm</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">nm</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:34_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:34"/><i><font color="Green">E54</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/>

<b>assume </b><a NAME="E3:34"/>
not  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:34"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:34"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:34"/>
<font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<b>hence </b><a NAME="E9:34"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div>
