<?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><br/>  for <font color="Olive" title="b4">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="b3">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Olive" title="b2">Prob</font> 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"><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="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b3">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b3">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">n</font></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><br/>  for <font color="Olive" title="b3">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="b2">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Olive" title="b1">Prob</font> 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"><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="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b2">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font></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><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">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> 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"><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="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b1">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></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">  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 <font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> 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"><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="prob_1.html#K2" title="PROB_1:func.2">Complement</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="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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></span><br/><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="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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/>









<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</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" 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="prob_1.html#K2" title="PROB_1:func.2">Complement</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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">means </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="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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> $1;<br/>
<a NAME="E2:26"/>
 <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_1.html#K2" title="PROB_1:func.2">Complement</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="E3:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</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_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E4:26"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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> <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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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_3.html#D1" title="SERIES_3:def.1">SERIES_3:def 1</a></span>;<br/>
<a NAME="E5:26"/><span class="lab"><font color="Green" title="E22">A4</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="#E3:26"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="prob_3.html#T21" title="PROB_3:th.21">PROB_3:21</a></span>;<br/>
<a NAME="E6:26"/><span class="lab"><font color="Green" title="E23">A5</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="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:26_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D2" title="PROB_1:def.2">PROB_1:def 2</a></span>;<br/>
<a NAME="E3:26_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="subset_1.html#D4" title="SUBSET_1:def.4">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E4:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">Omega</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="xboole_1.html#T50" title="XBOOLE_1:th.50">XBOOLE_1:50</a></span>;<br/>
<a NAME="E5:26_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<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="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">Omega</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E6:26_1"/><span class="lab"><font color="Green" title="E27">A9</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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p5">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></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="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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></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"><span class="p4">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p5">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</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="ref" href="prob_1.html#T33" title="PROB_1:th.33">PROB_1:33</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E7:26_1"/><span class="lab"><font color="Green" title="E28">A10</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="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></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"><span class="p4">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p5">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</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="#E4:26_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E5:26_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E6:26_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="prob_3.html#T21" title="PROB_3:th.21">PROB_3:21</a></span>;<br/>
<a NAME="E8:26_1"/>
 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">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="Olive" title="b1">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> 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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</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"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b1">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</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="26_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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">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" title="c7">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> 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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</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"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c7">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/><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" title="c7">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</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"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">A</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#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c7">A</font></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><br/>



<span class="kw">assume </span><a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</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" 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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">A</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#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c7">A</font></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><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">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="E3:26_1_1"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">n1</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="E5:26_1_1"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 ;<br/>
<a NAME="E6:26_1_1"/>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="#E5:26_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E7:26_1_1"/><span class="kw">then </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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">A</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="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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="c7">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="bor_cant.html#T6" target="_self" title="BOR_CANT:th.6">Th6</a>, <a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a></span>;<br/>
<a NAME="E8:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A14</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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">A</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#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p4">(<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="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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="c7">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p5">(<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>)</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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="prob_3.html#T21" title="PROB_3:th.21">PROB_3:21</a></span>;<br/>
<a NAME="E9:26_1_1"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="valued_0.html#K1" title="VALUED_0: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="prob_2.html#K1" title="PROB_2:func.1">.</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="c7">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<a NAME="E10:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A16</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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">A</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#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c7">A</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p4">(<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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="#E8:26_1_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="series_3.html#D1" title="SERIES_3:def.1">SERIES_3:def 1</a></span>;<br/>
<a NAME="E11:26_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="c7">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">k</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>  <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="E12:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A17</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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">A</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#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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"><font color="Maroon" title="c7">A</font> <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="txt" href="#E9:26_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E10:26_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E13:26_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="c7">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/>
<span class="kw">hence </span><a NAME="E14:26_1_1"/>
<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#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">A</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#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c7">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">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c7">A</font></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="txt" href="#E12:26_1_1"><span class="lab"><font color="Green" title="E35">A17</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="E9:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A18</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="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</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="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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> <span class="p5">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c4">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E7:26_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E10:26_1"/>
( <font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D4" title="SUBSET_1:def.4">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E11:26_1"/><span class="kw">then </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="c4">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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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"><span class="p2">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T20" title="PROB_1:th.20">PROB_1:20</a></span>;<br/>
<a NAME="E12:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A19</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="c4">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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <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_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T32" title="PROB_1:th.32">PROB_1:32</a></span>;<br/>
<a NAME="E13:26_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="c4">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="E14:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A20</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> <font color="Maroon" title="c4">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">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> 1 <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_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E15:26_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_1.html#K2" title="PROB_1:func.2">Complement</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="E16:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">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="E17:26_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="c4">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">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> 1 <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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">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="txt" href="#E16:26_1"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="txt" href="#E14:26_1"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="prob_1.html#D2" title="PROB_1:def.2">PROB_1:def 2</a></span>;<br/>
<a NAME="E18:26_1"/><span class="kw">then </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="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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_3.html#K1" title="SERIES_3:func.1">Partial_Product</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> <span class="p5">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</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_3.html#K1" title="SERIES_3:func.1">Partial_Product</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> <span class="p5">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">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"><span class="p3">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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> <span class="p5">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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="c6">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="txt" href="#E9:26_1"><span class="lab"><font color="Green" title="E29">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:26_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="ref" href="series_3.html#D1" title="SERIES_3:def.1">SERIES_3: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="E7:26"/>
 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="#E5:26"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E23">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:26"/>
<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="prob_1.html#K2" title="PROB_1:func.2">Complement</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</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"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
