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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>;<br/>

<b>assume </b><a NAME="E1:74"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:74_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D10" target="_self">Def10</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:74_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="measure6.html#K1">R_EAL</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:74"><i><font color="Green">E70</font></i></a>, <a class="ref" href="mesfunc5.html#D9" target="_self">Def9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E6:74_1"/>
(  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E7:74_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> &amp;  <a href="measure6.html#K1">R_EAL</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:74_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="mesfunc5.html#E5:74_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>hence </b><a NAME="E8:74_1"/>
contradiction
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:74"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:74"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><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="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D8" target="_self">Def8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1"><b>suppose </b></a><a NAME="E1:74_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:74_2_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><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="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:74"><i><font color="Green">E71</font></i></a></i>;<br/><div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:74_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:74_2_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><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="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:74_2_1"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E3:74_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/><a NAME="E4:74_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/><a NAME="E5:74_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E6:74_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><a NAME="E5:74_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:74_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:74_2_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:74"><i><font color="Green">E70</font></i></a>, <a class="ref" href="mesfunc5.html#D9" target="_self">Def9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E8:74_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E9:74_2_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:74_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="mesfunc5.html#E7:74_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E10:74_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_2"><b>suppose </b></a><a NAME="E1:74_2_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:74_2_2"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><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="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:74"><i><font color="Green">E71</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:74_2_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="measure6.html#K1">R_EAL</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:74"><i><font color="Green">E70</font></i></a>, <a class="ref" href="mesfunc5.html#D9" target="_self">Def9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E6:74_2_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E7:74_2_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><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="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:74_2_2"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E8:74_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> 1
 <b>by </b><i><a class="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/><a NAME="E9:74_2_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/><a NAME="E10:74_2_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:74_2_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/><b>hence </b><a NAME="E11:74_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:74_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="mesfunc5.html#E6:74_2_2"><i><font color="Green">E75</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_3"><b>suppose </b></a><a NAME="E1:74_2_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E2:74_2_3"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:74_2_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:74_2_3"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><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="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:74"><i><font color="Green">E71</font></i></a></i>;<br/><div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:74_2_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:74_2_3_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><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="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:74_2_3"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E3:74_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/><a NAME="E4:74_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/><a NAME="E5:74_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E6:74_2_3_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:74_2_3"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:74"><i><font color="Green">E70</font></i></a>, <a class="ref" href="mesfunc5.html#D9" target="_self">Def9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E8:74_2_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E9:74_2_3"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:74_2_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="mesfunc5.html#E7:74_2_3"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E10:74_2_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
