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

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon" title="c1">F</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 ;<br/>

<b>given </b><font color="Maroon" title="c2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:98"/><i><font color="Green" title="E74">A2</font></i>: 
<font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/>

<a NAME="E3:98"/><i><font color="Green" title="E75">A3</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:98_1"/><b>then </b><i><font color="Green" title="E76">A4</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:98"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="supinf_2.html#T63" target="_self" title="SUPINF_2:th.63">Th63</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">x</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> 0 as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<a NAME="E4:98_1"/><i><font color="Green" title="E77">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E5:98_1"/>
<font color="Maroon" title="c3">x</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:98_1"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="supinf_1.html#T33" title="SUPINF_1:th.33">SUPINF_1:33</a></i>;<br/>
<a NAME="E6:98_1"/><b>then </b>
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:98_1"><i><font color="Green" title="E76">A4</font></i></a>, <a class="txt" href="supinf_2.html#E4:98_1"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="supinf_1.html#T80" title="SUPINF_1:th.80">SUPINF_1:80</a></i>;<br/>
<b>hence </b><a NAME="E7:98_1"/>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:98"/>
(  ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E1:98_2"/><i><font color="Green" title="E76">A6</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c3">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:98_2"/><i><font color="Green" title="E77">A7</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:98_2_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:98"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="supinf_2.html#T59" target="_self" title="SUPINF_2:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E2:98_2_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c5">y</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c4">k</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<a NAME="E5:98_2"/><i><font color="Green" title="E78">A8</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63" target="_self" title="SUPINF_2:th.63">Th63</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<a NAME="E7:98_2"/><i><font color="Green" title="E79">A9</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:98"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="supinf_2.html#E1:98_2"><i><font color="Green" title="E76">A6</font></i></a>, <a class="txt" href="supinf_2.html#E3:98_2"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="supinf_2.html#E5:98_2"><i><font color="Green" title="E78">A8</font></i></a>, <a class="ref" href="supinf_2.html#D2" target="_self" title="SUPINF_2:def.2">Def2</a></i>;<br/>
<a NAME="E8:98_2"/><i><font color="Green" title="E80">A10</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E9:98_2"/>
<font color="Maroon" title="c6">x</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E7:98_2"><i><font color="Green" title="E79">A9</font></i></a>, <a class="ref" href="supinf_1.html#T33" title="SUPINF_1:th.33">SUPINF_1:33</a></i>;<br/>
<a NAME="E10:98_2"/><b>then </b>
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E7:98_2"><i><font color="Green" title="E79">A9</font></i></a>, <a class="txt" href="supinf_2.html#E8:98_2"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="supinf_1.html#T80" title="SUPINF_1:th.80">SUPINF_1:80</a></i>;<br/>
<b>hence </b><a NAME="E11:98_2"/>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:98"/>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="supinf_2.html#E3:98"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>


</div>
