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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> is <a href="int_1.html#V1">integer</a> );<br/>
<a NAME="E1:47"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is <a href="int_1.html#V1">integer</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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> 0</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">0 <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="irrat_1.html#T33" target="_self">Th33</a></i>
;<br/><b>hence </b><a NAME="E3:47_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="ref" href="irrat_1.html#T37" target="_self">Th37</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:47_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:47_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="int_1.html#V1">integer</a> )
 ;<br/><b>assume </b><a NAME="E2:47_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:47_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <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/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="irrat_1.html#E1:47_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="irrat_1.html#E2:47_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:47_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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"><font color="Maroon">c<sub>2</sub></font> <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="irrat_1.html#T33" target="_self">Th33</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="irrat_1.html#E2:47_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="irrat_1.html#T37" target="_self">Th37</a></i>;<br/><i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E8:47_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="int_1.html#NM1">Integer</a>
 ;<br/><b>hence </b><a NAME="E9:47_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="irrat_1.html#E7:47_1_2"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:47_1"/><b>then </b><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:47_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="irrat_1.html#E2:47_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="irrat_1.html#E4:47_1"><i><font color="Green">E43</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:47_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is <a href="int_1.html#V1">integer</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="series_1.html#K1">Partial_Sums</a> <a href="irrat_1.html#K5" target="_self">eseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="int_1.html#V1">integer</a>
 ;<br/>


</div>
