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

<b>assume </b><a NAME="E1:80"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 ;<br/>

<div><i><font color="Green">E37</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">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:80_1"/><i><font color="Green">E38</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">L</font></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:80_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</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="E4:80_1"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</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="E5:80_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">L</font></span>)</span>
 <b>by </b><i>, <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">k0</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k0</font> or  ex <font color="Olive">k0</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k0</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">E39</font></i>: 
for <font color="Olive">k0</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k0</font>
 ;<br/><div class="add"><div><i><font color="Green">E40</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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:80_2_1_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:80_2_1_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</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">K</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">K</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">n</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">n</font>
 ;<br/><a NAME="E2:80_2_1_2_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E3:80_2_1_2_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</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">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 ;<br/></div><b>end;</b></div><a NAME="E4:80_2_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a>
 <b>by </b><i/>;<br/><a NAME="E5:80_2_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</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">x</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">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:80_2_1_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:80_2_1_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:80_2_1_3"/>
<font color="Maroon">n</font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T17" target="_self">Th17</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:80_2_1_3"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a>, <a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><a NAME="E7:80_2_1"/><b>then </b><i><font color="Green">E47</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">L</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">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font> holds <br/> <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a>, <a class="ref" href="mesfunc5.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E10:80_2_1"/>
( <font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</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">L</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</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">k0</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k0</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">E94</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k0</font>
 ;<br/><b>reconsider </b><font color="Maroon">k0</font> = <font color="Maroon">k0</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<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">K</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">n</font> being  <a href="nat_1.html#NM1">Nat</a>  holds <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">K</font> or for <font color="Olive">K</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">K</font> or  ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st not <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">K</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">K</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">n</font> being  <a href="nat_1.html#NM1">Nat</a>  holds <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">K</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">K</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">E95</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  holds <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E4:80_2_2_1_1"/><i><font color="Green">E98</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">L</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="ref" href="mesfunc5.html#D4" target="_self">Def4</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">x</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">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:80_2_2_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:80_2_2_1_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:80_2_2_1_1_1"/>
<font color="Maroon">z</font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:80_2_2_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a>, <a class="ref" href="mesfunc5.html#T17" target="_self">Th17</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">K</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</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">E101</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">L</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</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">K</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">E102</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">L</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="ref" href="mesfunc5.html#T16" target="_self">Th16</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">E103</font></i>: 
for <font color="Olive">p</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">p</font> holds <br/> ex <font color="Olive">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</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">L</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">p</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">p</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">E104</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">p</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">L</font></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="mesfunc5.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</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">E106</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</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">L</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">e</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="measure6.html#T33">MEASURE6:33</a></i>;<br/>
<b>consider </b><font color="Maroon">x</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">E107</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">N0</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">N1</font>,<font color="Maroon">k0</font>;<br/>
<b>take </b>
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">N1</font>,<font color="Maroon">k0</font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</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">E108</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">N1</font>,<font color="Maroon">k0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:80_2_2_1_1_2_1"/>
<font color="Maroon">N1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">N1</font>,<font color="Maroon">k0</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">N1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E5:80_2_2_1_1_2_1"/><b>then </b><i><font color="Green">E109</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">L</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">e</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T20" target="_self">Th20</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">E121</font></i>: 
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D4" target="_self">Def4</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">L</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">L</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">e</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E8:80_2_2_1_1_2_1"/><b>then </b><i><font color="Green">E122</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">L</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">L</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">e</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">E123</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">L</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">L</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">e</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">e</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">L</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">L</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</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">L</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">L</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a>, <a class="ref" href="mesfunc5.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</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">e</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="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="mesfunc5.html#T24" target="_self">Th24</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">L</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">L</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">e</font></span>)</span>
 <b>by </b><i>, <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">E124</font></i>: 
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">L</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">e</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</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">L</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">e</font></span>)</span>
 <b>by </b><i>, <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">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</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">e</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">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">E125</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">e</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">L</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">L</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">e</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">L</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">p</font>
 <b>by </b><i>, <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">h</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">L</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="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="mesfunc5.html#T17" target="_self">Th17</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">E126</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">h</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">L</font></span>)</span>
 ;<br/><a NAME="E13:80_2_2_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T11" target="_self">Th11</a>, <a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a></i>;<br/><b>hence </b><a NAME="E14:80_2_2_1_1"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</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">L</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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</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">E128</font></i>: 
for <font color="Olive">K</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">K</font> or  ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st not <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">K</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">K</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">K</font>
 ;<br/><b>then consider </b><font color="Maroon">N0</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">E129</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D5" target="_self">Def5</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">n</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">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:80_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D3" target="_self">Def3</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">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</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">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 ;<br/></div><b>end;</b></div><a NAME="E3:80_2_2_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E4:80_2_2_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</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">E132</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">L</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">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k0</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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D4" target="_self">Def4</a></i>;<br/><b>then reconsider </b><font color="Maroon">h</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">L</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="mesfunc5.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>set </b><font color="Maroon">K</font> =  <a href="square_1.html#K2">max</a> 0,<font color="Maroon">h</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">h</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">E133</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">h</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">h</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> 0,<font color="Maroon">h</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">E134</font></i>: 
<font color="Maroon">h</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">h</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">N0</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">E135</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">h</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#D5" target="_self">Def5</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">L</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E11:80_2_2_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="mesfunc5.html#D4" target="_self">Def4</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">L</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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
