<?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="E1:80"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:80_1"/><i><font color="Green">E78</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/><a NAME="E2:80_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:80_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:80_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:80_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_2"/>
( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds not  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2_1"><b>suppose </b></a><a NAME="E1:80_2_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds not  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:80_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:80_2_1"><i><font color="Green">E78</font></i></a></i>;<br/><b>hence </b><a NAME="E2:80_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:80_2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:80_2_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:80_2_1_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:80_2_1"><i><font color="Green">E79</font></i></a></i>;<br/><b>hence </b><a NAME="E3:80_2_1_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&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="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:80_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><a NAME="E4:80_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E5:80_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self">Def11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:80_2_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:80_2_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:80_2_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:80_2_1"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:80_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/><a NAME="E8:80_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/><a NAME="E9:80_2_1"/><b>then </b>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a>, <a class="txt" href="mesfunc5.html#E7:80_2_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>hence </b><a NAME="E10:80_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:80_2_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="mesfunc5.html#E5:80_2_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2_2"><b>suppose </b></a><a NAME="E1:80_2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><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="E3:80_2_2"/><i><font color="Green">E78</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_2_2_1"/>
( ex <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 <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> or 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>  holds <br/>(  not 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st not <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2_2_1_1"><b>suppose </b></a><a NAME="E1:80_2_2_1_1"/>
ex <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 <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</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="E3:80_2_2_1_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <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#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:80_2_2_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:80_2_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="mesfunc5.html#E2:80"><i><font color="Green">E77</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:80_2_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:80_2_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:80_2_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:80_2_2_1_1"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:80_2_2_1_1"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/><a NAME="E7:80_2_2_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/><a NAME="E8:80_2_2_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E9:80_2_2_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:80_2_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E10:80_2_2_1_1"/><i><font color="Green">E83</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="supinf_1.html#K6">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:80_2_2_1_1_2"/><i><font color="Green">E84</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:80_2_2_1_1_2"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:80_2_2_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="mesfunc5.html#E9:80_2_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:80_2_2_1_1_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</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>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:80_2_2_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="measure6.html#T33">MEASURE6:33</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:80_2_2_1_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:80_2_2_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E7:80_2_2_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>take </b>
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:80_2_2_1_1_2_1"/><i><font color="Green">E87</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:80_2_2_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E3:80_2_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:80_2_2_1_1_2_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:80_2_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:80"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E5:80_2_2_1_1_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</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>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:80_2_2_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="mesfunc5.html#E5:80_2_2_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:80_2_2_1_1_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:80"><i><font color="Green">E77</font></i></a></i>;<br/><a NAME="E7:80_2_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</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> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</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>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a>, <a class="txt" href="mesfunc5.html#E1:80_2_2_1_1_2"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E8:80_2_2_1_1_2_1"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</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>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_2_2_1_1_2_1_1"/><i><font color="Green">E91</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</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>5</sub></font></span>)</span>
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:80_2_2_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a>, <a class="txt" href="mesfunc5.html#E9:80_2_2_1_1"><i><font color="Green">E82</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</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> 0</span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>
<br/>.= 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><b>hence </b><a NAME="E3:80_2_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:80_2_2_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="mesfunc5.html#E1:80_2_2_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>;<br/></div><b>end;</b></div><a NAME="E10:80_2_2_1_1_2_1"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</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>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:80_2_2_1_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><a NAME="E11:80_2_2_1_1_2_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</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>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:80_2_2_1_1_2_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:80_2_2_1_1_2_1"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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>10</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>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:80_2_2_1_1_2_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/><a NAME="E13:80_2_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <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>10</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>5</sub></font>
 <b>by </b><i><a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/><a NAME="E14:80_2_2_1_1_2_1"/><b>then </b>
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><a NAME="E15:80_2_2_1_1_2_1"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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>10</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T15">EXTREAL2:15</a></i>;<br/><a NAME="E16:80_2_2_1_1_2_1"/>
<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>10</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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>5</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:80_2_2_1_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/><b>hence </b><a NAME="E17:80_2_2_1_1_2_1"/>
<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>10</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:80_2_2_1_1_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="extreal2.html#T59">EXTREAL2:59</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E4:80_2_2_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="mesfunc5.html#E9:80_2_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E12:80_2_2_1_1"/><i><font color="Green">E84</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E13:80_2_2_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D8" target="_self">Def8</a>, <a class="txt" href="mesfunc5.html#E10:80_2_2_1_1"><i><font color="Green">E83</font></i></a></i>;<br/><b>hence </b><a NAME="E14:80_2_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self">Def11</a></i>;<br/><b>hence </b><a NAME="E15:80_2_2_1_1"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:80_2_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="mesfunc5.html#E12:80_2_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="mesfunc5.html#E13:80_2_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2_2_1_2"><b>suppose </b></a><a NAME="E1:80_2_2_1_2"/><i><font color="Green">E79</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>  holds <br/>(  not 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st not <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:80_2_2_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<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="E3:80_2_2_1_2_1"/><i><font color="Green">E80</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:80_2_2_1_2"><i><font color="Green">E79</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_2_1_2_1_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="E1:80_2_2_1_2_1_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="E2:80_2_2_1_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#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:80"><i><font color="Green">E76</font></i></a></i>;<br/><b>hence </b><a NAME="E3:80_2_2_1_2_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:80_2_2_1_2_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:80_2_2_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <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>2</sub></font>
 ;<br/></div><b>end;</b></div><a NAME="E3:80_2_2_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" target="_self">Def9</a></i>;<br/><b>hence </b><a NAME="E4:80_2_2_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self">Def11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_2_2_1_2_2"/><i><font color="Green">E82</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><a NAME="E2:80_2_2_1_2_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:80"><i><font color="Green">E77</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E3:80_2_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="mesfunc5.html#E1:80_2_2_1_2_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="square_1.html#K2">max</a> 0,<font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E4:80_2_2_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> 0,<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="E5:80_2_2_1_2_2"/><b>then </b><i><font color="Green">E83</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> 0,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E6:80_2_2_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> 0,<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="E7:80_2_2_1_2_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> 0,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><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="E9:80_2_2_1_2_2"/><i><font color="Green">E85</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> 0,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</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="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:80_2_2_1_2_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="mesfunc5.html#E1:80_2_2_1_2"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E10:80_2_2_1_2_2"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E7:80_2_2_1_2_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="mesfunc5.html#E9:80_2_2_1_2_2"><i><font color="Green">E85</font></i></a></i>;<br/><b>hence </b><a NAME="E11:80_2_2_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:80"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:80_2_2_1_2"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:80_2_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="mesfunc5.html#E3:80_2_2_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
