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



<b>assume </b><a NAME="E1:48"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" 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="E3:48"/><i><font color="Green">E42</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</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="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green">E45</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a>, <a class="ref" href="square_1.html#T2">SQUARE_1:2</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="hidden.html#M1">set</a>  = <font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">seq</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:48"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<i><font color="Green">E48</font></i>: <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/>
<a NAME="E8:48"/><i><font color="Green">E49</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E9:48"/><i><font color="Green">E50</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="irrat_1.html#E6:48"><i><font color="Green">E46</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="irrat_1.html#E4:48"><i><font color="Green">E45</font></i></a>, <a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E11:48"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">seq</font> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="irrat_1.html#E6:48"><i><font color="Green">E46</font></i></a>, <a class="txt" href="irrat_1.html#E8:48"><i><font color="Green">E49</font></i></a>, <a class="ref" href="series_1.html#T29">SERIES_1:29</a></i>;<br/>
<a NAME="E12:48"/><i><font color="Green">E76</font></i>: 
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="irrat_1.html#E7:48"><i><font color="Green">E48</font></i></a>, <a class="txt" href="irrat_1.html#E9:48"><i><font color="Green">E50</font></i></a>, <a class="txt" href="irrat_1.html#E10:48"><i><font color="Green">E51</font></i></a>, <a class="ref" href="series_1.html#T29">SERIES_1:29</a></i>;<br/>
<a NAME="E13:48"/><i><font color="Green">E77</font></i>: 
<a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#T15">SERIES_1:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><i><font color="Green">E78</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seq_1.html#K2">.</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="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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="newton.html#K12">!</a> </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"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/><b>hence </b><a NAME="E2:48_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E3:48_3"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:48_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="txt" href="irrat_1.html#E2:48"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:48"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq</font>
 
<b>by </b><i><a class="txt" href="irrat_1.html#E11:48"><i><font color="Green">E52</font></i></a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E16:48"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="series_1.html#T13">SERIES_1:13</a></i>;<br/>


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