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

<b>let </b><font color="Maroon">Omega</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Sigma</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">FSeq</font> be    <a href="prob_3.html#M1" target="_self">FinSequence</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">ASeq</font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</font>;<br/>









<b>assume </b><a NAME="E1:67"/><i><font color="Green">E36</font></i>: 
( ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> holds <br/><font color="Maroon">ASeq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSeq</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Olive">k</font> ) &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st not <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> holds <br/><font color="Maroon">ASeq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 ;<br/>

<a NAME="E2:67"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="prob_3.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E3:67"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="prob_2.html#T11">PROB_2:11</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="prob_3.html#T6" target="_self">Th6</a>, <a class="ref" href="prob_1.html#T64">PROB_1:64</a></i>
;<br/>

<b>thus </b><a NAME="E2:67_1"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">m1</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/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E2:67_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <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="E3:67_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:67_1_2"/><b>then </b>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:67_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">ASeq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="prob_3.html#T5" target="_self">Th5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> = 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prob_2.html#T11">PROB_2:11</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="prob_3.html#E4:67"><i><font color="Green">E59</font></i></a>, <a class="ref" href="prob_1.html#T64">PROB_1:64</a></i>
;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E4:67"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:67_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font>
 ;<br/>

<a NAME="E2:67_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="prob_2.html#T11">PROB_2:11</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">FSeq</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#T7" target="_self">Th7</a>, <a class="ref" href="prob_3.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="prob_3.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:67"/>
1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<b>then </b><i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</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">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</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">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> 1
<b>by </b><i/>
;<br/>
<a NAME="E7:67"/><i><font color="Green">E76</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 implies ( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:67_4"/><b>then </b><i><font color="Green">E77</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:67_4"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="prob_3.html#T7" target="_self">Th7</a>, , <a class="txt" href="prob_3.html#E4:67"><i><font color="Green">E59</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:67_4"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:67_4_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> )
 ;<br/>

<a NAME="E2:67_4_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T12">FINSEQ_3:12</a></i>;<br/>
<a NAME="E3:67_4_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m1</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/>
<b>thus </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="prob_3.html#E4:67"><i><font color="Green">E59</font></i></a></i>
;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 0
<b>by </b><i/>
;<br/>
<a NAME="E9:67"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E10:67"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:67_6"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</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/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">k1</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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:67_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="prob_3.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:67"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<a NAME="E12:67"/><i><font color="Green">E131</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_7"><b>proof </b></a><div class="add">

<a NAME="E1:67_7"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_7_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:67_7_1"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:67_7_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>thus </b><a NAME="E4:67_7_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="prob_3.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:67_7"/><b>then </b>
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:67_7"/>
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> )
 <b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:67"/><i><font color="Green">E134</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_8_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_1"><b>suppose </b></a><a NAME="E1:67_8_1_1_1"/><i><font color="Green">E135</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:67_8_1_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="prob_3.html#E2:67"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E3:67_8_1_1_1"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a>, <a class="ref" href="prob_3.html#D1" target="_self">Def1</a>, </i>;<br/><b>hence </b><a NAME="E4:67_8_1_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a>, <a class="txt" href="prob_3.html#E3:67"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2"><b>suppose </b></a><a NAME="E1:67_8_1_1_2"/><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:67_8_1_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prob_3.html#E2:67"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E3:67_8_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:67_8_1_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:67"><i><font color="Green">E52</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">seq1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:67_8_1_1_2"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">seq1</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> holds <br/><font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> ) &amp;  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> ) &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> ) );<br/><a NAME="E7:67_8_1_1_2"/>
 ex <font color="Olive">seq</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( ( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">seq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Olive">seq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> ) &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Olive">seq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_8_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:67_8_1_1_2_1"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_8_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:67_8_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <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="67_8_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_8_1_1_2_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or ( <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#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> ) or ( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:67_8_1_1_2_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_8_1_1_2_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,0]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:67_8_1_1_2_1_1_1_1_2"/>
( <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#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_8_1_1_2_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">seq1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:67_8_1_1_2_1_1_1_1_3"/>
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_8_1_1_2_1_1_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:67_8_1_1_2_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:67_8_1_1_2_1"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:67_8_1_1_2_1"/><i><font color="Green">E145</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="ref" href="prob_3.html#T13" target="_self">Th13</a>, <a class="ref" href="prob_3.html#T12" target="_self">Th12</a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_8_1_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:67_8_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E2:67_8_1_1_2_1_2"/><b>then </b>
 ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> ) &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="prob_3.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E3:67_8_1_1_2_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is <a href="xreal_0.html#V1">real</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="prob_3.html#T14" target="_self">Th14</a>, <a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/>
<b>take </b><font color="Maroon">seq</font> = <font color="Maroon">f</font>;<br/>

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

<a NAME="E7:67_8_1_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <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="E8:67_8_1_1_2_1"/><b>then </b>
 ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">seq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">seq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">seq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="prob_3.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E9:67_8_1_1_2_1"/>
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">seq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <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#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">seq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> ) &amp; ( <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#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">seq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> ) )
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">seq2</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E9:67_8_1_1_2"/><i><font color="Green">E147</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( ( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">seq2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">seq2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> ) &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> implies <font color="Maroon">seq2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> ) )
 ;<br/><a NAME="E10:67_8_1_1_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">seq2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_3.html#T11" target="_self">Th11</a>, <a class="ref" href="prob_3.html#D2" target="_self">Def2</a>, <a class="ref" href="prob_3.html#T12" target="_self">Th12</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">seq2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><font color="Maroon">seq2</font> <a href="funct_2.html#K3">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> 0
<b>by </b><i><a class="ref" href="prob_3.html#T12" target="_self">Th12</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
;<br/><a NAME="E12:67_8_1_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 ;<br/><a NAME="E13:67_8_1_1_2"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_8_1_1_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:67_8_1_1_2_3"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_8_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_8_1_1_2_3_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 ) or ( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:67_8_1_1_2_3_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_8_1_1_2_3_1_1_1"/>
<font color="Maroon">seq2</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="prob_3.html#T8" target="_self">Th8</a>, <a class="ref" href="prob_3.html#T11" target="_self">Th11</a>, <a class="ref" href="prob_3.html#T10" target="_self">Th10</a>, <a class="ref" href="prob_3.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:67_8_1_1_2_3_1_1_2"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 ;<br/><div class="add"><a NAME="E2:67_8_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:67_8_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:67"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E4:67_8_1_1_2_3_1_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="prob_3.html#E7:67"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><font color="Maroon">seq2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E7:67"><i><font color="Green">E76</font></i></a>, , <a class="ref" href="prob_3.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E6:67"><i><font color="Green">E75</font></i></a>, <a class="txt" href="prob_3.html#E7:67"><i><font color="Green">E76</font></i></a>, , <a class="ref" href="prob_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="txt" href="prob_3.html#E7:67"><i><font color="Green">E76</font></i></a>, , <a class="ref" href="prob_3.html#T8" target="_self">Th8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_8_1_1_2_3_1_1_3"><b>case </b></a><a NAME="E1:67_8_1_1_2_3_1_1_3"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 ;<br/><div class="add"><a NAME="E2:67_8_1_1_2_3_1_1_3"/><b>then </b><i><font color="Green">E155</font></i>: 
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:67_8_1_1_2_3_1_1_3"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <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>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:67_8_1_1_2_3_1_1_3"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>thus </b><font color="Maroon">seq2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E7:67"><i><font color="Green">E76</font></i></a>, <a class="ref" href="prob_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E2:67"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="prob_3.html#E2:67"><i><font color="Green">E52</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="E3:67_8_1_1_2_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E14:67_8_1_1_2"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="prob_3.html#T14" target="_self">Th14</a>, );<br/></i><b>hence </b> <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E2:67"><i><font color="Green">E52</font></i></a>, <a class="ref" href="prob_3.html#D1" target="_self">Def1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:67_8"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:67"/><i><font color="Green">E157</font></i>: 
 <a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font> is <a href="prob_2.html#V1">disjoint_valued</a>
 
<b>by </b><i/>;<br/>
<a NAME="E15:67"/><i><font color="Green">E158</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="txt" href="prob_3.html#E1:67"><i><font color="Green">E36</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_9"><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="E1:67_9"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:67_9"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E17:67"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="ref" href="series_1.html#T17">SERIES_1:17</a></i>;<br/>
<a NAME="E18:67"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a>, <a class="ref" href="prob_3.html#D2" target="_self">Def2</a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E19:67"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E20:67"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E21:67"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1">Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6" target="_self">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E22:67"/>
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FSeq</font></span>)</span> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1">Union</a> <font color="Maroon">ASeq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> &amp;  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> )
 <b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a>, , </i>;<br/>


</div>
