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<b>let </b><font color="Maroon">F1</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> , <font color="Maroon">F2</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="E1:95"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">F1</font> is <a href="supinf_2.html#V6" target="_self">nonnegative</a>
 ;<br/>

<b>assume </b><a NAME="E2:95"/><i><font color="Green">E28</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">F1</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F2</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E3:95"/>
<font color="Maroon">F1</font> is    <a href="supinf_2.html#M1" target="_self">Num</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <font color="Maroon">F1</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">F1</font> as    <a href="supinf_2.html#M1" target="_self">Num</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <font color="Maroon">F1</font> ;<br/>
<a NAME="E5:95"/><i><font color="Green">E29</font></i>: 
 <a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a>  <a href="supinf_2.html#K14" target="_self">Ser</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" target="_self">rng</a> <font color="Maroon">F1</font></span>)</span>,<font color="Maroon">N1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:95"/>
<font color="Maroon">F2</font> is    <a href="supinf_2.html#M1" target="_self">Num</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <font color="Maroon">F2</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">N2</font> = <font color="Maroon">F2</font> as    <a href="supinf_2.html#M1" target="_self">Num</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <font color="Maroon">F2</font> ;<br/>
<a NAME="E8:95"/><i><font color="Green">E30</font></i>: 
 <a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a>  <a href="supinf_2.html#K14" target="_self">Ser</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" target="_self">rng</a> <font color="Maroon">F2</font></span>)</span>,<font color="Maroon">N2</font>
 
<b>by </b><i/>;<br/>
<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/>

<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><span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F1</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</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" target="_self">Ser</a> <font color="Maroon">F2</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E9:95"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<a NAME="E1:95_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F1</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="supinf_2.html#K13" target="_self">.</a> 0
 
<b>by </b><i><a class="txt" href="supinf_2.html#E27"><i><font color="Green">Lemma71</font></i></a>, </i>;<br/>
<a NAME="E2:95_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F2</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="supinf_2.html#K13" target="_self">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>hence </b><a NAME="E3:95_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:95"/><i><font color="Green">E51</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="95_2"><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="E1:95_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">F1</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</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" target="_self">Ser</a> <font color="Maroon">F2</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:95_2"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F1</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">F1</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><a class="txt" href="supinf_2.html#E27"><i><font color="Green">Lemma71</font></i></a>, </i>;<br/>
<a NAME="E3:95_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F2</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F2</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">F2</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><a class="ref" href="supinf_2.html#T21" target="_self">Th21</a>, </i>;<br/>
<a NAME="E4:95_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">F1</font> <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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F2</font> <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>
 
<b>by </b><i/>;<br/>
<a NAME="E5:95_2"/><i><font color="Green">E56</font></i>: 
 <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">F1</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#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:95_2"/>
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F1</font> <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>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E7:95_2"/>
<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="ref" href="supinf_2.html#D5" target="_self">Def5</a>, , , , , <a class="ref" href="supinf_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:95"/>
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>
<b>hence </b><a NAME="E12:95"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F1</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</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" target="_self">Ser</a> <font color="Maroon">F2</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 ;<br/>


</div>
