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

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green">E32</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E3:35"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<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> , <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span> ) );<br/>
<a NAME="E4:35"/><i><font color="Green">E34</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>   ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><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="E1:35_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:35_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:35_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="series_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:35_1"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Olive">r</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">s1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:35"/><i><font color="Green">E36</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>,<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1">RCOMP_1:sch 1</a>(<a class="txt" href="series_1.html#E1"><i><font color="Green">Lemma29</font></i></a>);<br/></i>
<a NAME="E7:35"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="series_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:35"/><i><font color="Green">E38</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1"><b>suppose </b></a><a NAME="E1:35_2_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:35_2_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</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> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="series_1.html#E2:35"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E3:35_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="txt" href="series_1.html#E3:35"><i><font color="Green">E33</font></i></a>, <a class="ref" href="prepower.html#T99">PREPOWER:99</a></i>;<br/><a NAME="E4:35_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E5:35_2_1_1_1"/>
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E1:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="series_1.html#E3:35"><i><font color="Green">E33</font></i></a>, <a class="txt" href="series_1.html#E4:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_2.html#T164">REAL_2:164</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_2"><b>suppose </b></a><a NAME="E1:35_2_1_1_2"/>
 ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:35_2_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</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="E5:35_2_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="series_1.html#E3:35"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E6:35_2_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:35_2_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:35_2_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E4:35"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E9:35_2_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E10:35_2_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E7:35"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E2:35"><i><font color="Green">E32</font></i></a>, <a class="txt" href="series_1.html#E6:35"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E2:35"><i><font color="Green">E32</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T101">XCMPLX_1:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font>
<b>by </b><i><a class="txt" href="series_1.html#E7:35"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T36">POWER:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><a NAME="E12:35_2_1_1_2"/>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E8:35"><i><font color="Green">E38</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E13:35_2_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E2:35"><i><font color="Green">E32</font></i></a>, </i>;<br/><a NAME="E14:35_2_1_1_2"/>
<font color="Maroon">n</font> <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="E15:35_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="series_1.html#E7:35"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_2.html#T143">REAL_2:143</a></i>;<br/><a NAME="E16:35_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> 0
 <b>by </b><i>, , , <a class="ref" href="prepower.html#T98">PREPOWER:98</a></i>;<br/><a NAME="E17:35_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="prepower.html#T85">PREPOWER:85</a></i>;<br/><b>hence </b><a NAME="E18:35_2_1_1_2"/>
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E3:35_2_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="real_2.html#T118">REAL_2:118</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:35_2"/>
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E10:35"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">s1</font> is <a href="seqm_3.html#V4">non-increasing</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D14">SEQM_3:def 14</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E12:35"/><i><font color="Green">E74</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">s2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>now </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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_3_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1"><b>suppose </b></a><a NAME="E1:35_3_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_3_1_1_1"/>
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E1:35"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_2"><b>suppose </b></a><a NAME="E1:35_3_1_1_2"/>
 ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:35_3_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</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="E5:35_3_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="series_1.html#E6:35"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E6:35_3_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:35_3_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E7:35"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E8:35_3_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E9:35_3_1_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>hence </b><a NAME="E10:35_3_1_1_2"/>
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E2:35"><i><font color="Green">E32</font></i></a>, <a class="txt" href="series_1.html#E8:35"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:35_3"/>
( <font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="series_1.html#E4:35"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">s3</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E15:35"/><i><font color="Green">E84</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">s3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4"><b>now </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="E1:35_4"/>
2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E2:35_4"/><b>then </b><i><font color="Green">E86</font></i>: 
2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>thus </b><i><font color="Green">E87</font></i>: <font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E4:35"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E2:35"><i><font color="Green">E32</font></i></a>, <a class="txt" href="series_1.html#E8:35"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" target="_self">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T33">POWER:33</a></i>
<br/>.= 
2 <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
2 <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>

;<br/><b>hence </b><a NAME="E4:35_4"/>
<font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E5:35_4"/>
<font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="power.html#T38">POWER:38</a></i>;<br/><b>hence </b><a NAME="E6:35_4"/>
<font color="Maroon">s3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="power.html#K1">-root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="series_1.html#E6:35"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_5"><b>now </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="E1:35_5"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E2:35_5"/><i><font color="Green">E90</font></i>: 
( 2 <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 2 <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">s3</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">s3</font> <a href="funct_2.html#K3">.</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="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K1">-root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="power.html#K3">to_power</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="txt" href="series_1.html#E7:35"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K1">-root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="newton.html#K2">|^</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="power.html#T46">POWER:46</a></i>
<br/>.= 
2 <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="power.html#T5">POWER:5</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E18:35"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">s3</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E19:35"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">s3</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<a NAME="E20:35"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">s3</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s3</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">s3</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> 0
<b>by </b><i>, <a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>
<br/>.= 
2 <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E8:35"><i><font color="Green">E38</font></i></a></i>
;<br/>
<a NAME="E22:35"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> 2 <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/>
<a NAME="E23:35"/>
( 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 
<b>by </b><i>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<a NAME="E24:35"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i>, <a class="ref" href="power.html#T41">POWER:41</a></i>;<br/>
<a NAME="E25:35"/><b>then </b>
<font color="Maroon">s2</font> is <a href="series_1.html#V1" target="_self">summable</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E6:35"><i><font color="Green">E36</font></i></a>, <a class="txt" href="series_1.html#E7:35"><i><font color="Green">E37</font></i></a>, , </i>;<br/>
<a NAME="E26:35"/><b>then </b>
<font color="Maroon">s1</font> is <a href="series_1.html#V1" target="_self">summable</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E3:35"><i><font color="Green">E33</font></i></a>, , </i>;<br/>
<a NAME="E27:35"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">s1</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="series_1.html#V1" target="_self">summable</a>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_7"><b>now </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="E1:35_7"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><i><font color="Green">E99</font></i>: <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</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="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</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="txt" href="series_1.html#E2:35"><i><font color="Green">E32</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
;<br/><a NAME="E3:35_7"/>
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:35_7"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="series_1.html#T2" target="_self">Th2</a>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_8"><b>now </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/><b>assume </b><a NAME="E1:35_8"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E2:35_8"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</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="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</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="txt" href="series_1.html#E2:35"><i><font color="Green">E32</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
;<br/><b>hence </b><a NAME="E4:35_8"/>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E30:35"/><b>then </b>
<font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="series_1.html#V1" target="_self">summable</a>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E31:35"/>
<font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a>
 <b>by </b><i/>;<br/>


</div>
