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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> )</span><br/>



<b>thus </b><a NAME="E1:34"/>
(  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> implies  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><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" title="E31">A1</font></i>: 
 <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font></span><br/>

<b>assume </b><a NAME="E2:34_1"/><i><font color="Green" title="E32">A2</font></i>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:34_1"/>
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">nm</font> = <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">m</font> as    <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>  <b>by </b><i><a class="txt" href="catalan2.html#E2:34_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>set </b><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1;<br/>
<a NAME="E5:34_1"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">nm</font> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</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="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">nm</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="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">nm</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E2:34_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green" title="E10">Lm2</font></i></a>, <a class="txt" href="catalan2.html#E13"><i><font color="Green" title="E12">Lm4</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:34_1"/><b>then </b>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">nm</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T9" target="_self" title="CATALAN2:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E7:34_1"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E1:34_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:34"/><i><font color="Green" title="E31">A3</font></i>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/>

<b>assume </b><a NAME="E3:34"/>
not  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:34"/><i><font color="Green" title="E32">A4</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>,<font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">p</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="E7:34"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">p</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">p</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E5:34"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></i>;<br/>
<a NAME="E8:34"/>
<font color="Maroon" title="c4">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E7:34"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>;<br/>
<b>hence </b><a NAME="E9:34"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E2:34"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="catalan2.html#E7:34"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="catalan2.html#T6" target="_self" title="CATALAN2:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
