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

<span class="kw">let </span><font color="Maroon" title="c1">Omega</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Sigma</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b2">Prob</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Olive" title="b1">Sigma</font><br/>  for <font color="Olive" title="b3">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">Sigma</font>  st  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b3">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  holds <br/>( <font color="Olive" title="b2">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K8" title="BOR_CANT:func.8">@lim_sup</a> <font color="Olive" title="b3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Olive" title="b2">Prob</font>,<font color="Olive" title="b3">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Olive" title="b2">Prob</font>,<font color="Olive" title="b3">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Prob</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b2">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  holds <br/>( <font color="Olive" title="b1">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K8" title="BOR_CANT:func.8">@lim_sup</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Olive" title="b1">Prob</font>,<font color="Olive" title="b2">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Olive" title="b1">Prob</font>,<font color="Olive" title="b2">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">Prob</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  holds <br/>( <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K8" title="BOR_CANT:func.8">@lim_sup</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Olive" title="b1">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Olive" title="b1">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  implies ( <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K8" title="BOR_CANT:func.8">@lim_sup</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:29"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K8" title="BOR_CANT:func.8">@lim_sup</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/>

<a NAME="E2:29"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></span>;<br/>
<a NAME="E3:29"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<a NAME="E1:29_1"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:29_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:29_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:29"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T29" title="PROB_3:th.29">PROB_3:29</a></span>;<br/>
<a NAME="E5:29"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 <a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T27" title="PROB_3:th.27">PROB_3:27</a></span>;<br/>
<a NAME="E6:29"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
(  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</a></span>;<br/>
<a NAME="E7:29"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b3">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b3">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">n</font>, <font color="Maroon" title="c7">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E1:29_2"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>;<br/>
<a NAME="E2:29_2"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:29_2"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E4:29_2"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D2" title="PROB_3:def.2">PROB_3:def 2</a></span>;<br/>
<a NAME="E5:29_2"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:29_2"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_2"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E4:29_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E3:29_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E1:29_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E7:29_2"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_2_1"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:29_2_1"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:29_2_1"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T39" title="PROB_1:th.39">PROB_1:39</a></span>;<br/>
<a NAME="E4:29_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:29_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E6:29_2_1"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
( <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/>
<a NAME="E7:29_2_1"/><span class="lab"><font color="Green" title="E41">A23</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp; <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:29_2_1"/><span class="lab"><font color="Green" title="E42">A24</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font> implies <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/>
<a NAME="E9:29_2_1"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_2_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E5:29_2_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E6:29_2_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="txt" href="#E2:29_2_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="txt" href="#E1:29_2_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E7:29_2_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="txt" href="#E8:29_2_1"><span class="lab"><font color="Green" title="E42">A24</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="prob_3.html#D2" title="PROB_3:def.2">PROB_3:def 2</a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></span>;<br/>
<a NAME="E10:29_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:29_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_2_1"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:29_2"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E7:29_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:29_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:29"/><span class="lab"><font color="Green" title="E29">A26</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">k</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> </span><br/>

<a NAME="E1:29_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_3"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:29"/><span class="lab"><font color="Green" title="E30">A27</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span><br/>



<a NAME="E1:29_4"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font></span><br/>

<a NAME="E1:29_4_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:29_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E3:29_4_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:29_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A29</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E5:29_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:29_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_4_1"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="txt" href="#E4:29_4_1"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:29_4"/>
<font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:29"/><span class="lab"><font color="Green" title="E31">A30</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> </span><br/>

<a NAME="E1:29_5"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29"><span class="lab"><font color="Green" title="E30">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_5"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29"><span class="lab"><font color="Green" title="E29">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:29"/><span class="lab"><font color="Green" title="E32">A31</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E1:29_6"/><span class="lab"><font color="Green" title="E33">A32</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E28">A8</font></span></a></span>;<br/>
<a NAME="E2:29_6"/><span class="lab"><font color="Green" title="E34">A33</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T41" title="PROB_3:th.41">PROB_3:41</a></span>;<br/>
<a NAME="E3:29_6"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29"><span class="lab"><font color="Green" title="E31">A30</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:29_6"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_6"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E1:29_6"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="ref" href="seq_2.html#T18" title="SEQ_2:th.18">SEQ_2:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:29"/><span class="lab"><font color="Green" title="E33">A34</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">Prob</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Prob</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E1:29_7"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E32">A31</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_7"/>
<font color="Maroon" title="c3">Prob</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T41" title="PROB_3:th.41">PROB_3:41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:29"/><span class="lab"><font color="Green" title="E34">A35</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">Prob</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Prob</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:29_8"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_8"/>
<font color="Maroon" title="c3">Prob</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E33">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:29"/><span class="lab"><font color="Green" title="E35">A36</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<a NAME="E1:29_9"/><span class="lab"><font color="Green" title="E36">A37</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:29_9"/><span class="lab"><font color="Green" title="E37">A38</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_9"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E3:29_9"/><span class="lab"><font color="Green" title="E38">A39</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:29"><span class="lab"><font color="Green" title="E34">A35</font></span></a></span>;<br/>
<a NAME="E4:29_9"/>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bor_cant.html#D11" target="_self" title="BOR_CANT:def.11">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:29_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bor_cant.html#D7" target="_self" title="BOR_CANT:def.7">Def9</a>, <a class="txt" href="#E2:29_9"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E3:29_9"><span class="lab"><font color="Green" title="E38">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:29"/><span class="lab"><font color="Green" title="E36">A40</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="seq_2.html#T17" title="SEQ_2:th.17">SEQ_2:17</a></span>;<br/>
<a NAME="E16:29"/><span class="lab"><font color="Green" title="E37">A41</font></span>: 
(  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  implies  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:29_10"/><span class="lab"><font color="Green" title="E38">A42</font></span>: 
 <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span></span><br/>

<a NAME="E2:29_10"/><span class="lab"><font color="Green" title="E39">A43</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<a NAME="E1:29_10_1"/><span class="lab"><font color="Green" title="E40">A44</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T34" title="PROB_1:th.34">PROB_1:34</a>, <a class="ref" href="prob_3.html#T23" title="PROB_3:th.23">PROB_3:23</a></span>;<br/>
<a NAME="E2:29_10_1"/><span class="lab"><font color="Green" title="E41">A45</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:29_10_1"/><span class="lab"><font color="Green" title="E42">A46</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:29_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_10_1"><span class="lab"><font color="Green" title="E42">A46</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:29_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_10_1"><span class="lab"><font color="Green" title="E41">A45</font></span></a>, <a class="txt" href="#E1:29_10_1"><span class="lab"><font color="Green" title="E40">A44</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:29_10"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_10_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:29_10_2"/><span class="lab"><font color="Green" title="E40">A47</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_10_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<a NAME="E1:29_10_2_1"/><span class="lab"><font color="Green" title="E41">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_10"><span class="lab"><font color="Green" title="E39">A43</font></span></a></span>;<br/>
<a NAME="E2:29_10_2_1"/><span class="lab"><font color="Green" title="E42">A49</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29"><span class="lab"><font color="Green" title="E35">A36</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:29_10_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_10_2_1"><span class="lab"><font color="Green" title="E41">A48</font></span></a>, <a class="txt" href="#E2:29_10_2_1"><span class="lab"><font color="Green" title="E42">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E2:29_10_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E1:29_10"><span class="lab"><font color="Green" title="E38">A42</font></span></a>, <a class="txt" href="#E1:29_10_2"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="ref" href="seq_2.html#T18" title="SEQ_2:th.18">SEQ_2:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:29_10"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="bor_cant.html#K7" title="BOR_CANT:func.7">Union_Shift_Seq</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:29"/><span class="lab"><font color="Green" title="E38">A50</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Olive" title="b1">A</font>)</span>)</span> &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Olive" title="b1">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_11"/><span class="lab"><font color="Green" title="E39">A51</font></span>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/>

<a NAME="E2:29_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A52</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></span>;<br/>
<a NAME="E3:29_11"/><span class="lab"><font color="Green" title="E41">A53</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span><br/>

<a NAME="E1:29_11_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_11"><span class="lab"><font color="Green" title="E40">A52</font></span></a>, <a class="ref" href="series_1.html#T15" title="SERIES_1:th.15">SERIES_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_11_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:29_11"/><span class="lab"><font color="Green" title="E42">A54</font></span>: 
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font>, <font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:29_11_2"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:29_11_2"/><span class="lab"><font color="Green" title="E43">A55</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_11"><span class="lab"><font color="Green" title="E41">A53</font></span></a></span>;<br/>
<a NAME="E3:29_11_2"/><span class="lab"><font color="Green" title="E44">A56</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_11"><span class="lab"><font color="Green" title="E41">A53</font></span></a>, <a class="txt" href="#E2:29_11_2"><span class="lab"><font color="Green" title="E43">A55</font></span></a></span>;<br/>
<a NAME="E4:29_11_2"/><span class="lab"><font color="Green" title="E45">A57</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_11_2"><span class="lab"><font color="Green" title="E44">A56</font></span></a></span>;<br/>
<a NAME="E5:29_11_2"/><span class="lab"><font color="Green" title="E46">A58</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span><br/>



<a NAME="E1:29_11_2_1"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E1:29_11_2_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:29_11_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A59</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E3:29_11_2_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:29_11_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A60</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E5:29_11_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:29_11_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_11_2_1_1"><span class="lab"><font color="Green" title="E47">A59</font></span></a>, <a class="txt" href="#E4:29_11_2_1_1"><span class="lab"><font color="Green" title="E48">A60</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:29_11_2_1"/>
<font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:29_11_2"/><span class="lab"><font color="Green" title="E47">A61</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E1:29_11_2_2"/><span class="lab"><font color="Green" title="E48">A62</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<a NAME="E2:29_11_2_2"/>
<span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bor_cant.html#D11" target="_self" title="BOR_CANT:def.11">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:29_11_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_11_2"><span class="lab"><font color="Green" title="E46">A58</font></span></a>, <a class="txt" href="#E1:29_11_2_2"><span class="lab"><font color="Green" title="E48">A62</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:29_11_2"/><span class="lab"><font color="Green" title="E48">A64</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_2_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:29_11_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_11_2"><span class="lab"><font color="Green" title="E45">A57</font></span></a>, <a class="txt" href="#E6:29_11_2"><span class="lab"><font color="Green" title="E47">A61</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_11_2_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_11_2"><span class="lab"><font color="Green" title="E47">A61</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:29_11_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_2_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_11_2_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_11_2_4_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_11_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_11_2_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:29_11_2_4_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_11_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_11_2_4_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:29_11_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A65</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:29_11_2_4_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_11_2_4_1_2"><span class="lab"><font color="Green" title="E49">A65</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:29_11_2_4_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_11_2_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:29_11_2_4_1_3"/><span class="lab"><font color="Green" title="E49">A66</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:29_11_2_4_1_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_11_2_4_1_3"><span class="lab"><font color="Green" title="E49">A66</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:29_11_2_4_1_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:29_11_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_11_2"><span class="lab"><font color="Green" title="E48">A64</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:29_11"/><span class="lab"><font color="Green" title="E43">A67</font></span>: 
( (  for <font color="Olive" title="b1">sr</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">sr</font> holds <br/> ex <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">sr</font> ) implies  for <font color="Olive" title="b1">sr</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">sr</font> holds <br/> ex <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">sr</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:29_11_3"/><span class="lab"><font color="Green" title="E44">A68</font></span>: 
 for <font color="Olive" title="b1">sr</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">sr</font> holds <br/> ex <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">sr</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">sr</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">sr</font> holds <br/> ex <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">sr</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">sr</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font> implies  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:29_11_3"/><span class="lab"><font color="Green" title="E45">A69</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:29_11_3"/><span class="lab"><font color="Green" title="E46">A70</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_11_3"><span class="lab"><font color="Green" title="E44">A68</font></span></a>, <a class="txt" href="#E2:29_11_3"><span class="lab"><font color="Green" title="E45">A69</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:29_11_3"/><span class="lab"><font color="Green" title="E47">A71</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font></span><br/>

<a NAME="E6:29_11_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_11_3"><span class="lab"><font color="Green" title="E47">A71</font></span></a>, <a class="txt" href="#E4:29_11"><span class="lab"><font color="Green" title="E42">A54</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:29_11_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">sr</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_11_3"><span class="lab"><font color="Green" title="E46">A70</font></span></a>, <a class="txt" href="#E5:29_11_3"><span class="lab"><font color="Green" title="E47">A71</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:29_11"/><span class="lab"><font color="Green" title="E44">A72</font></span>: 
(  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1 is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_11"><span class="lab"><font color="Green" title="E39">A51</font></span></a>, <a class="txt" href="#E5:29_11"><span class="lab"><font color="Green" title="E43">A67</font></span></a>, <a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></span>;<br/>
<a NAME="E7:29_11"/><span class="lab"><font color="Green" title="E45">A73</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">B</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E9:29_11"/><span class="lab"><font color="Green" title="E46">A74</font></span>: 
<font color="Maroon" title="c6">B</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">B1</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>;<br/>
<a NAME="E10:29_11"/><span class="lab"><font color="Green" title="E47">A75</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_11"><span class="lab"><font color="Green" title="E39">A51</font></span></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></span>;<br/>
<a NAME="E11:29_11"/><span class="lab"><font color="Green" title="E48">A76</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span><br/>

<a NAME="E1:29_11_4"/><span class="lab"><font color="Green" title="E49">A77</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E1:29_11_4_1"/><span class="lab"><font color="Green" title="E50">A78</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<a NAME="E2:29_11_4_1"/><span class="lab"><font color="Green" title="E51">A79</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">A</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span><br/>



<a NAME="E1:29_11_4_1_1"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_4_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span><br/>

<a NAME="E1:29_11_4_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:29_11_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A80</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E3:29_11_4_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:29_11_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A81</font></span>: 
<font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E5:29_11_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:29_11_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_11_4_1_1_1"><span class="lab"><font color="Green" title="E52">A80</font></span></a>, <a class="txt" href="#E4:29_11_4_1_1_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:29_11_4_1_1"/>
<font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c11">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:29_11_4_1"/>
<span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bor_cant.html#D11" target="_self" title="BOR_CANT:def.11">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:29_11_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_11_4_1"><span class="lab"><font color="Green" title="E50">A78</font></span></a>, <a class="txt" href="#E2:29_11_4_1"><span class="lab"><font color="Green" title="E51">A79</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:29_11_4"/><span class="lab"><font color="Green" title="E50">A82</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_11_4"><span class="lab"><font color="Green" title="E49">A77</font></span></a></span>;<br/>
<a NAME="E3:29_11_4"/>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_11"><span class="lab"><font color="Green" title="E47">A75</font></span></a>, <a class="ref" href="series_1.html#T15" title="SERIES_1:th.15">SERIES_1:15</a></span>;<br/>
<a NAME="E4:29_11_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_11_4"><span class="lab"><font color="Green" title="E50">A82</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:29_11_4"/>
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_11"><span class="lab"><font color="Green" title="E45">A73</font></span></a>, <a class="txt" href="#E9:29_11"><span class="lab"><font color="Green" title="E46">A74</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:29_11"/><span class="lab"><font color="Green" title="E49">A83</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:29_11_5"/>
 ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span><br/>

<span class="kw">thus </span><a NAME="E1:29_11_5_1"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_11"><span class="lab"><font color="Green" title="E48">A76</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:29_11_5"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T19" title="SEQ_4:th.19">SEQ_4:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E50">A84</font></span>:  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29_11"><span class="lab"><font color="Green" title="E49">A83</font></span></a>, <a class="ref" href="seq_4.html#T26" title="SEQ_4:th.26">SEQ_4:26</a></span>
;<br/>
<a NAME="E14:29_11"/><span class="lab"><font color="Green" title="E51">A85</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_11"><span class="lab"><font color="Green" title="E46">A74</font></span></a>, <a class="txt" href="#E6:29_11"><span class="lab"><font color="Green" title="E44">A72</font></span></a>, <a class="ref" href="seq_2.html#T6" title="SEQ_2:th.6">SEQ_2:6</a></span>;<br/>
<a NAME="E15:29_11"/>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:29_11"><span class="lab"><font color="Green" title="E50">A84</font></span></a>, <a class="txt" href="#E14:29_11"><span class="lab"><font color="Green" title="E51">A85</font></span></a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:29_11"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>)</span>)</span> &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c5">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_11"><span class="lab"><font color="Green" title="E44">A72</font></span></a>, <a class="ref" href="seq_4.html#T21" title="SEQ_4:th.21">SEQ_4:21</a>, <a class="ref" href="seq_4.html#T22" title="SEQ_4:th.22">SEQ_4:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:29"/>
(  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E17:29"><span class="lab"><font color="Green" title="E38">A50</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:29"/>
( <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K8" title="BOR_CANT:func.8">@lim_sup</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="bor_cant.html#K11" title="BOR_CANT:func.11">Sum_Shift_Seq</a> (<font color="Maroon" title="c3">Prob</font>,<font color="Maroon" title="c4">A</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E15:29"><span class="lab"><font color="Green" title="E36">A40</font></span></a>, <a class="txt" href="#E16:29"><span class="lab"><font color="Green" title="E37">A41</font></span></a>, <a class="ref" href="prob_2.html#D1" title="PROB_2:def.1">PROB_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
