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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Olive" title="b1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Olive" title="b1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Olive" title="b1">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#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Olive" title="b1">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="Olive" title="b1">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b1">p</font></span>)</span></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"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></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"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green" title="E10">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></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"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> )</span><br/>

<b>set </b><font color="Maroon" title="c4">ap</font> = <font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font>;<br/>
<b>set </b><font color="Maroon" title="c5">bp</font> = <font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font>;<br/>
<b>set </b><font color="Maroon" title="c6">ab</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font>;<br/>
<div><i><font color="Green" title="E13">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font> )</span><br/><i><font color="Green" title="E14">A5</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self" title="LP_SPACE:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
;<br/><b>thus </b><a NAME="E2:9_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self" title="LP_SPACE:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font> )</span><br/><b>thus </b><a NAME="E3:9_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self" title="LP_SPACE:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font></span><br/><b>thus </b><a NAME="E4:9_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="lp_space.html#E1:9_1"><i><font color="Green" title="E14">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_2_1"/>
( <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="lp_space.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1"><b>case </b></a><a NAME="E1:9_2_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></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"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )</span><br/><div class="add"><b>hence </b><a NAME="E2:9_2_1_1"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></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"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 <b>by </b><i><a class="txt" href="lp_space.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="lp_space.html#E3:9"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="lp_space.html#E4:9"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="holder_1.html#T13" title="HOLDER_1:th.13">HOLDER_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2"><b>case </b></a><a NAME="E1:9_2_1_2"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></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"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )</span><br/><div class="add"><b>hence </b><a NAME="E2:9_2_1_2"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></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"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 <b>by </b><i><a class="txt" href="lp_space.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="lp_space.html#E3:9"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="lp_space.html#E8"><i><font color="Green" title="E8">Lm4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:9"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">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="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></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"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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