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<div class="add">

<b>let </b><font color="Maroon" title="c1">F1</font>, <font color="Maroon" title="c2">F2</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="E1:96"/><i><font color="Green" title="E70">A1</font></i>: 
<font color="Maroon" title="c1">F1</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 ;<br/>

<b>assume </b><a NAME="E2:96"/><i><font color="Green" title="E71">A2</font></i>: 
for <font color="Olive" title="b1">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>  holds  <font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">N1</font> = <font color="Maroon" title="c1">F1</font> as    <a href="supinf_2.html#M1" title="SUPINF_2:mode.1">Num</a> of  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c1">F1</font> <b>by </b><i><a class="ref" href="supinf_2.html#D16" target="_self" title="SUPINF_2:def.16">Def16</a></i>;<br/>
<a NAME="E4:96"/><i><font color="Green" title="E72">A3</font></i>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c1">F1</font></span>)</span>,<font color="Maroon" title="c3">N1</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D21" target="_self" title="SUPINF_2:def.21">Def21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">N2</font> = <font color="Maroon" title="c2">F2</font> as    <a href="supinf_2.html#M1" title="SUPINF_2:mode.1">Num</a> of  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c2">F2</font> <b>by </b><i><a class="ref" href="supinf_2.html#D16" target="_self" title="SUPINF_2:def.16">Def16</a></i>;<br/>
<a NAME="E6:96"/><i><font color="Green" title="E73">A4</font></i>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c2">F2</font></span>)</span>,<font color="Maroon" title="c4">N2</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D21" target="_self" title="SUPINF_2:def.21">Def21</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">n</font> be    <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> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b><span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> $1;<br/>
<a NAME="E7:96"/><i><font color="Green" title="E74">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<a NAME="E1:96_1"/><i><font color="Green" title="E75">A6</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0
 
<b>by </b><i><a class="txt" href="supinf_2.html#E4:96"><i><font color="Green" title="E72">A3</font></i></a>, <a class="ref" href="supinf_2.html#D17" target="_self" title="SUPINF_2:def.17">Def17</a></i>;<br/>
<a NAME="E2:96_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0
 
<b>by </b><i><a class="txt" href="supinf_2.html#E6:96"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="supinf_2.html#D17" target="_self" title="SUPINF_2:def.17">Def17</a></i>;<br/>
<b>hence </b><a NAME="E3:96_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:96"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="supinf_2.html#E1:96_1"><i><font color="Green" title="E75">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:96"/><i><font color="Green" title="E75">A7</font></i>: 
for <font color="Olive" title="b1">k</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:96_2"/><i><font color="Green" title="E76">A8</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">k</font>
 ;<br/>

<a NAME="E2:96_2"/><i><font color="Green" title="E77">A9</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="txt" href="supinf_2.html#E4:96"><i><font color="Green" title="E72">A3</font></i></a>, <a class="ref" href="supinf_2.html#D17" target="_self" title="SUPINF_2:def.17">Def17</a></i>;<br/>
<a NAME="E3:96_2"/><i><font color="Green" title="E78">A10</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="txt" href="supinf_2.html#E6:96"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="supinf_2.html#D17" target="_self" title="SUPINF_2:def.17">Def17</a></i>;<br/>
<a NAME="E4:96_2"/><i><font color="Green" title="E79">A11</font></i>: 
<font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:96"><i><font color="Green" title="E71">A2</font></i></a></i>;<br/>
<a NAME="E5:96_2"/><i><font color="Green" title="E80">A12</font></i>: 
 <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#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:96"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="supinf_2.html#T59" target="_self" title="SUPINF_2:th.59">Th59</a></i>;<br/>
<a NAME="E6:96_2"/>
 <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> <font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:96"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="supinf_2.html#T58" target="_self" title="SUPINF_2:th.58">Th58</a></i>;<br/>
<b>hence </b><a NAME="E7:96_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:96_2"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="supinf_2.html#E2:96_2"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="supinf_2.html#E3:96_2"><i><font color="Green" title="E78">A10</font></i></a>, <a class="txt" href="supinf_2.html#E4:96_2"><i><font color="Green" title="E79">A11</font></i></a>, <a class="txt" href="supinf_2.html#E5:96_2"><i><font color="Green" title="E80">A12</font></i></a>, <a class="txt" href="supinf_2.html#E28"><i><font color="Green" title="E24">Lm4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:96"/>
for <font color="Olive" title="b1">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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="supinf_2.html#E7:96"><i><font color="Green" title="E74">A5</font></i></a>, <a class="txt" href="supinf_2.html#E8:96"><i><font color="Green" title="E75">A7</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:96"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">n</font>
 ;<br/>


</div>
