<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  for <font color="Olive" title="b1">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  for <font color="Olive" title="b2">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="Olive" title="b1">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/>( <font color="Olive" title="b1">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp; <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:16"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  for <font color="Olive" title="b2">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="Olive" title="b1">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/>( <font color="Olive" title="b1">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp; <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E2:16"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies ( <font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp; <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:16"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 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="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp; <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E4:16"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 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  <span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E1:16_1"/>
<font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/>
<a NAME="E2:16_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T2" title="ABSVALUE:th.2">ABSVALUE:2</a></span>;<br/>
<a NAME="E3:16_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:16_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lp_space.html#D1" target="_self" title="LP_SPACE:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:16"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
 for <font color="Olive" title="b1">m</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  <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><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> ] <span class="kw">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">m</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
 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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_2_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:16_2"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>;<br/>
<a NAME="E3:16_2"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:16_2"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a>);<br/></span>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">m</font>

<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:16"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
 for <font color="Olive" title="b1">e</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> holds <br/> ex <font color="Olive" title="b2">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>  st <br/> for <font color="Olive" title="b3">m</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="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">e</font> implies  ex <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>  st <br/> for <font color="Olive" title="b2">m</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_3"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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>  st <br/> for <font color="Olive" title="b2">m</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">e</font></span><br/>

<span class="kw">take </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">e</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">m</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:16_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">e</font></span><br/>

 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:16_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E23">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></span>;<br/>
<a NAME="E8:16"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/>
<a NAME="E9:16"/><span class="kw">then </span>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:16"/>
( <font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp; <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
