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<div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">F</font> is <a href="supinf_2.html#V6" target="_self">nonnegative</a>
 ;<br/>

<b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:98"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E3:98"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 implies  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+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">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:98_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0 as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:98_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E5:98_1"/>
<font color="Maroon">x</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D5" target="_self">Def5</a>, <a class="ref" href="supinf_1.html#T33">SUPINF_1:33</a></i>;<br/>
<a NAME="E6:98_1"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" target="_self">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="supinf_1.html#T80">SUPINF_1:80</a></i>;<br/>
<b>hence </b><a NAME="E7:98_1"/>
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:98"/>
(  ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1 implies  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+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">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:98_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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="E3:98_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-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" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T21" target="_self">Th21</a>, <a class="ref" href="supinf_2.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E2:98_2_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font> <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="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E5:98_2"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E7:98_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#D5" target="_self">Def5</a>, , , </i>;<br/>
<a NAME="E8:98_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E9:98_2"/>
<font color="Maroon">x</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T33">SUPINF_1:33</a></i>;<br/>
<a NAME="E10:98_2"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" target="_self">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, , <a class="ref" href="supinf_1.html#T80">SUPINF_1:80</a></i>;<br/>
<b>hence </b><a NAME="E11:98_2"/>
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:98"/>
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>


</div>
