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<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:79"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 ;<br/>

<b>consider </b><font color="Maroon">sumf</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E3:79"/><i><font color="Green">E37</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sumf</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">sumf</font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">sumf</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sumf</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> implies 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sumf</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:79"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:79"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </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="E1:79_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>assume </b><a NAME="E2:79_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">n</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="E4:79_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:79_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">sumf</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sumf</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"><font color="Maroon">F</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E6:79_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:79_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E8:79_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:79_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sumf</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a>, <a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E10:79_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sumf</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="extreal2.html#T7">EXTREAL2:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:79"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">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#S2">NAT_1:sch 2</a>(<a class="ref" href="mesfunc5.html#D4" target="_self">Def4</a>, );<br/></i>
<b>hence </b><a NAME="E7:79"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D3" target="_self">Def3</a></i>;<br/>


</div>
