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<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="E1:30"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:30"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E3:30"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font>, <font color="Maroon">y</font> = <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">z</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</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="E3:30_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E4:30_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:30"><i><font color="Green">E19</font></i></a>, <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a>, <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="measure6.html#E3:30"><i><font color="Green">E22</font></i></a>, <a class="ref" href="supinf_2.html#T20">SUPINF_2:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:30"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E5:30"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:30_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">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="E2:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</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> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/>
<b>hence </b><a NAME="E3:30_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E6:30"/>
( <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E3:30"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<a NAME="E7:30"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>
<a NAME="E8:30"/><b>then </b>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:30"><i><font color="Green">E19</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a>, <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E9:30"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:30"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


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