let Omega be non empty set ; :: thesis: for Sigma being SigmaField of Omega
for P being Probability of Sigma
for FSeq being FinSequence of Sigma
for ASeq being SetSequence of Sigma st ( for k being Nat st k in dom FSeq holds
ASeq . k = FSeq . k ) & ( for k being Nat st not k in dom FSeq holds
ASeq . k = {} ) holds
( Partial_Sums (P * ASeq) is convergent & Sum (P * ASeq) = (Partial_Sums (P * ASeq)) . (len FSeq) & P . (Union ASeq) <= Sum (P * ASeq) & Sum (P * FSeq) = Sum (P * ASeq) )
let Sigma be SigmaField of Omega; :: thesis: for P being Probability of Sigma
for FSeq being FinSequence of Sigma
for ASeq being SetSequence of Sigma st ( for k being Nat st k in dom FSeq holds
ASeq . k = FSeq . k ) & ( for k being Nat st not k in dom FSeq holds
ASeq . k = {} ) holds
( Partial_Sums (P * ASeq) is convergent & Sum (P * ASeq) = (Partial_Sums (P * ASeq)) . (len FSeq) & P . (Union ASeq) <= Sum (P * ASeq) & Sum (P * FSeq) = Sum (P * ASeq) )
let P be Probability of Sigma; :: thesis: for FSeq being FinSequence of Sigma
for ASeq being SetSequence of Sigma st ( for k being Nat st k in dom FSeq holds
ASeq . k = FSeq . k ) & ( for k being Nat st not k in dom FSeq holds
ASeq . k = {} ) holds
( Partial_Sums (P * ASeq) is convergent & Sum (P * ASeq) = (Partial_Sums (P * ASeq)) . (len FSeq) & P . (Union ASeq) <= Sum (P * ASeq) & Sum (P * FSeq) = Sum (P * ASeq) )
let FSeq be FinSequence of Sigma; :: thesis: for ASeq being SetSequence of Sigma st ( for k being Nat st k in dom FSeq holds
ASeq . k = FSeq . k ) & ( for k being Nat st not k in dom FSeq holds
ASeq . k = {} ) holds
( Partial_Sums (P * ASeq) is convergent & Sum (P * ASeq) = (Partial_Sums (P * ASeq)) . (len FSeq) & P . (Union ASeq) <= Sum (P * ASeq) & Sum (P * FSeq) = Sum (P * ASeq) )
let ASeq be SetSequence of Sigma; :: thesis: ( ( for k being Nat st k in dom FSeq holds
ASeq . k = FSeq . k ) & ( for k being Nat st not k in dom FSeq holds
ASeq . k = {} ) implies ( Partial_Sums (P * ASeq) is convergent & Sum (P * ASeq) = (Partial_Sums (P * ASeq)) . (len FSeq) & P . (Union ASeq) <= Sum (P * ASeq) & Sum (P * FSeq) = Sum (P * ASeq) ) )
assume that
A1: for k being Nat st k in dom FSeq holds
ASeq . k = FSeq . k and
A2: for k being Nat st not k in dom FSeq holds
ASeq . k = {} ; :: thesis: ( Partial_Sums (P * ASeq) is convergent & Sum (P * ASeq) = (Partial_Sums (P * ASeq)) . (len FSeq) & P . (Union ASeq) <= Sum (P * ASeq) & Sum (P * FSeq) = Sum (P * ASeq) )
A3: ASeq . 0 = {} by A1, A2, Th55;
A4: (P * ASeq) . 0 = P . (ASeq . 0) by FUNCT_2:15
.= 0 by A3, VALUED_0:def 19 ;
A5: for k being Nat st k in dom FSeq holds
(P * ASeq) . k = (P * FSeq) . k
proof
let k be Nat; :: thesis: ( k in dom FSeq implies (P * ASeq) . k = (P * FSeq) . k )
assume A6: k in dom FSeq ; :: thesis: (P * ASeq) . k = (P * FSeq) . k
k in NAT by ORDINAL1:def 12;
hence (P * ASeq) . k = P . (ASeq . k) by FUNCT_2:15
.= P . (FSeq . k) by A1, A6
.= (P * FSeq) . k by A6, FUNCT_1:13 ;
:: thesis: verum
end;
1 = 0 + 1 ;
then A7: (Partial_Sums (P * ASeq)) . 1 = ((Partial_Sums (P * ASeq)) . 0) + ((P * ASeq) . 1) by SERIES_1:def 1
.= ((P * ASeq) . 0) + ((P * ASeq) . 1) by SERIES_1:def 1
.= (P * ASeq) . 1 by A4 ;
A8: ( len FSeq >= 1 implies ( (Partial_Sums (P * ASeq)) . 1 = (P * FSeq) . 1 & ( for m being Nat st m <> 0 & m < len FSeq holds
(Partial_Sums (P * ASeq)) . (m + 1) = ((Partial_Sums (P * ASeq)) . m) + ((P * FSeq) . (m + 1)) ) ) )
proof
assume len FSeq >= 1 ; :: thesis: ( (Partial_Sums (P * ASeq)) . 1 = (P * FSeq) . 1 & ( for m being Nat st m <> 0 & m < len FSeq holds
(Partial_Sums (P * ASeq)) . (m + 1) = ((Partial_Sums (P * ASeq)) . m) + ((P * FSeq) . (m + 1)) ) )
then 1 in dom FSeq by Th2;
hence (Partial_Sums (P * ASeq)) . 1 = (P * FSeq) . 1 by A5, A7; :: thesis: for m being Nat st m <> 0 & m < len FSeq holds
(Partial_Sums (P * ASeq)) . (m + 1) = ((Partial_Sums (P * ASeq)) . m) + ((P * FSeq) . (m + 1))
thus for m being Nat st m <> 0 & m < len FSeq holds
(Partial_Sums (P * ASeq)) . (m + 1) = ((Partial_Sums (P * ASeq)) . m) + ((P * FSeq) . (m + 1)) :: thesis: verum
proof
let m be Nat; :: thesis: ( m <> 0 & m < len FSeq implies (Partial_Sums (P * ASeq)) . (m + 1) = ((Partial_Sums (P * ASeq)) . m) + ((P * FSeq) . (m + 1)) )
assume that
m <> 0 and
A9: m < len FSeq ; :: thesis: (Partial_Sums (P * ASeq)) . (m + 1) = ((Partial_Sums (P * ASeq)) . m) + ((P * FSeq) . (m + 1))
reconsider m1 = m as Element of NAT by ORDINAL1:def 12;
m + 1 in Seg (len FSeq) by A9, FINSEQ_3:11;
then A10: m + 1 in dom FSeq by FINSEQ_1:def 3;
thus (Partial_Sums (P * ASeq)) . (m + 1) = ((Partial_Sums (P * ASeq)) . m1) + ((P * ASeq) . (m1 + 1)) by SERIES_1:def 1
.= ((Partial_Sums (P * ASeq)) . m) + ((P * FSeq) . (m + 1)) by A5, A10 ; :: thesis: verum
end;
end;
defpred S1[ Nat] means (Partial_Sums (P * ASeq)) . (((len FSeq) + 1) + $1) = (Partial_Sums (P * ASeq)) . (len FSeq);
A11: for m being Nat holds (P * ASeq) . (((len FSeq) + 1) + m) = 0
proof
set k = (len FSeq) + 1;
let m be Nat; :: thesis: (P * ASeq) . (((len FSeq) + 1) + m) = 0
reconsider m1 = m as Element of NAT by ORDINAL1:def 12;
((len FSeq) + 1) + m >= (len FSeq) + 1 by NAT_1:11;
then ((len FSeq) + 1) + m > len FSeq by Lm1;
then not ((len FSeq) + 1) + m in dom FSeq by FINSEQ_3:25;
then A12: ASeq . (((len FSeq) + 1) + m) = {} by A2;
thus (P * ASeq) . (((len FSeq) + 1) + m) = P . (ASeq . (((len FSeq) + 1) + m1)) by FUNCT_2:15
.= 0 by A12, VALUED_0:def 19 ; :: thesis: verum
end;
A13: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume A14: (Partial_Sums (P * ASeq)) . (((len FSeq) + 1) + k) = (Partial_Sums (P * ASeq)) . (len FSeq) ; :: thesis: S1[k + 1]
reconsider k1 = k as Element of NAT by ORDINAL1:def 12;
(Partial_Sums (P * ASeq)) . ((((len FSeq) + 1) + k) + 1) = ((Partial_Sums (P * ASeq)) . (((len FSeq) + 1) + k1)) + ((P * ASeq) . (((len FSeq) + 1) + (k1 + 1))) by SERIES_1:def 1
.= ((Partial_Sums (P * ASeq)) . (((len FSeq) + 1) + k)) + 0 by A11 ;
hence S1[k + 1] by A14; :: thesis: verum
end;
now :: thesis: for n being Element of NAT holds (P * (Partial_Diff_Union ASeq)) . n <= (P * ASeq) . n
let n be Element of NAT ; :: thesis: (P * (Partial_Diff_Union ASeq)) . n <= (P * ASeq) . n
(Partial_Diff_Union ASeq) . n c= ASeq . n by Th17;
hence (P * (Partial_Diff_Union ASeq)) . n <= (P * ASeq) . n by Th5; :: thesis: verum
end;
then A15: for n being Element of NAT holds (Partial_Sums (P * (Partial_Diff_Union ASeq))) . n <= (Partial_Sums (P * ASeq)) . n by SERIES_1:14;
A16: Partial_Sums (P * (Partial_Diff_Union ASeq)) is convergent by Th45;
(Partial_Sums (P * ASeq)) . (((len FSeq) + 1) + 0) = ((Partial_Sums (P * ASeq)) . (len FSeq)) + ((P * ASeq) . (((len FSeq) + 1) + 0)) by SERIES_1:def 1
.= ((Partial_Sums (P * ASeq)) . (len FSeq)) + 0 by A11 ;
then A17: S1[ 0 ] ;
A18: for k being Nat holds S1[k] from NAT_1:sch 2(A17, A13);
 A19: for m being Nat st (len FSeq) + 1 <= m holds
(Partial_Sums (P * ASeq)) . m = (Partial_Sums (P * ASeq)) . (len FSeq)
proof
let m be Nat; :: thesis: ( (len FSeq) + 1 <= m implies (Partial_Sums (P * ASeq)) . m = (Partial_Sums (P * ASeq)) . (len FSeq) )
assume (len FSeq) + 1 <= m ; :: thesis: (Partial_Sums (P * ASeq)) . m = (Partial_Sums (P * ASeq)) . (len FSeq)
then ex k being Nat st m = ((len FSeq) + 1) + k by NAT_1:10;
hence (Partial_Sums (P * ASeq)) . m = (Partial_Sums (P * ASeq)) . (len FSeq) by A18; :: thesis: verum
end;
then A20: lim (Partial_Sums (P * ASeq)) = (Partial_Sums (P * ASeq)) . (len FSeq) by Th3;
then A21: Sum (P * ASeq) = (Partial_Sums (P * ASeq)) . (len FSeq) by SERIES_1:def 3;
A22: Sum (P * FSeq) = Sum (P * ASeq)
proof
now :: thesis: Sum (P * FSeq) = Sum (P * ASeq)
per cases ( len FSeq = 0 or len FSeq <> 0 ) ;
suppose len FSeq = 0 ; :: thesis: Sum (P * FSeq) = Sum (P * ASeq)
then ( len (P * FSeq) = 0 & Sum (P * ASeq) = 0 ) by A4, A21, Th61, SERIES_1:def 1;
hence Sum (P * FSeq) = Sum (P * ASeq) by Th62; :: thesis: verum
end;
suppose A23: len FSeq <> 0 ; :: thesis: Sum (P * FSeq) = Sum (P * ASeq)
then 1 <= len FSeq by NAT_1:14;
then A24: 1 <= len (P * FSeq) by Th61;
then consider seq1 being Real_Sequence such that
A25: seq1 . 1 = (P * FSeq) . 1 and
A26: for n being Nat st 0 <> n & n < len (P * FSeq) holds
seq1 . (n + 1) = (seq1 . n) + ((P * FSeq) . (n + 1)) and
A27: Sum (P * FSeq) = seq1 . (len (P * FSeq)) by Th63;
defpred S2[ set , set ] means ex n being Nat st
( n = $1 & ( n = 0 implies $2 = 0 ) & ( n <> 0 & n <= len (P * FSeq) implies $2 = seq1 . n ) & ( n <> 0 & n > len (P * FSeq) implies $2 = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) ) );
ex seq being Real_Sequence st
for n being Nat holds
( ( n = 0 implies seq . n = 0 ) & ( n <> 0 & n <= len (P * FSeq) implies seq . n = seq1 . n ) & ( n <> 0 & n > len (P * FSeq) implies seq . n = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) ) )
proof
A28: for x being set st x in NAT holds
ex y being set st S2[x,y]
proof
let x be set ; :: thesis: ( x in NAT implies ex y being set st S2[x,y] )
assume x in NAT ; :: thesis: ex y being set st S2[x,y]
then reconsider n = x as Element of NAT ;
now :: thesis: ( ( n = 0 & S2[x, 0 ] ) or ( n <> 0 & n <= len (P * FSeq) & S2[x,seq1 . n] ) or ( n <> 0 & not n <= len (P * FSeq) & S2[x,(Partial_Sums (P * ASeq)) . (len (P * FSeq))] ) )
per cases ( n = 0 or ( n <> 0 & n <= len (P * FSeq) ) or ( n <> 0 & not n <= len (P * FSeq) ) ) ;
case n = 0 ; :: thesis: S2[x, 0 ]
hence S2[x, 0 ] ; :: thesis: verum
end;
case ( n <> 0 & n <= len (P * FSeq) ) ; :: thesis: S2[x,seq1 . n]
hence S2[x,seq1 . n] ; :: thesis: verum
end;
case ( n <> 0 & not n <= len (P * FSeq) ) ; :: thesis: S2[x,(Partial_Sums (P * ASeq)) . (len (P * FSeq))]
hence S2[x,(Partial_Sums (P * ASeq)) . (len (P * FSeq))] ; :: thesis: verum
end;
end;
end;
hence ex y being set st S2[x,y] ; :: thesis: verum
end;
consider f being Function such that
A29: ( dom f = NAT & ( for x being set st x in NAT holds
S2[x,f . x] ) ) from CLASSES1:sch 1(A28);
now :: thesis: for x being set st x in NAT holds
f . x is real
let x be set ; :: thesis: ( x in NAT implies f . x is real )
assume x in NAT ; :: thesis: f . x is real
then ex n being Nat st
( n = x & ( n = 0 implies f . x = 0 ) & ( n <> 0 & n <= len (P * FSeq) implies f . x = seq1 . n ) & ( n <> 0 & n > len (P * FSeq) implies f . x = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) ) ) by A29;
hence f . x is real ; :: thesis: verum
end;
then reconsider f = f as Real_Sequence by A29, SEQ_1:1;
take seq = f; :: thesis: for n being Nat holds
( ( n = 0 implies seq . n = 0 ) & ( n <> 0 & n <= len (P * FSeq) implies seq . n = seq1 . n ) & ( n <> 0 & n > len (P * FSeq) implies seq . n = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) ) )
let n be Nat; :: thesis: ( ( n = 0 implies seq . n = 0 ) & ( n <> 0 & n <= len (P * FSeq) implies seq . n = seq1 . n ) & ( n <> 0 & n > len (P * FSeq) implies seq . n = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) ) )
n in NAT by ORDINAL1:def 12;
then ex k being Nat st
( k = n & ( k = 0 implies seq . n = 0 ) & ( k <> 0 & k <= len (P * FSeq) implies seq . n = seq1 . k ) & ( k <> 0 & k > len (P * FSeq) implies seq . n = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) ) ) by A29;
hence ( ( n = 0 implies seq . n = 0 ) & ( n <> 0 & n <= len (P * FSeq) implies seq . n = seq1 . n ) & ( n <> 0 & n > len (P * FSeq) implies seq . n = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) ) ) ; :: thesis: verum
end;
then consider seq2 being Real_Sequence such that
A30: for n being Nat holds
( ( n = 0 implies seq2 . n = 0 ) & ( n <> 0 & n <= len (P * FSeq) implies seq2 . n = seq1 . n ) & ( n <> 0 & n > len (P * FSeq) implies seq2 . n = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) ) ) ;
defpred S3[ Nat] means seq2 . $1 = (Partial_Sums (P * ASeq)) . $1;
A31: for k being Nat st S3[k] holds
S3[k + 1]
proof
let k be Nat; :: thesis: ( S3[k] implies S3[k + 1] )
assume A32: S3[k] ; :: thesis: S3[k + 1]
now :: thesis: ( ( k = 0 & seq2 . 1 = (Partial_Sums (P * ASeq)) . 1 ) or ( k <> 0 & k <= (len (P * FSeq)) - 1 & seq2 . (k + 1) = (Partial_Sums (P * ASeq)) . (k + 1) ) or ( k <> 0 & not k <= (len (P * FSeq)) - 1 & seq2 . (k + 1) = (Partial_Sums (P * ASeq)) . (k + 1) ) )
per cases ( k = 0 or ( k <> 0 & k <= (len (P * FSeq)) - 1 ) or ( k <> 0 & not k <= (len (P * FSeq)) - 1 ) ) ;
case k = 0 ; :: thesis: seq2 . 1 = (Partial_Sums (P * ASeq)) . 1
thus seq2 . 1 = (Partial_Sums (P * ASeq)) . 1 by A8, A23, A24, A25, A30, NAT_1:14; :: thesis: verum
end;
case A33: ( k <> 0 & k <= (len (P * FSeq)) - 1 ) ; :: thesis: seq2 . (k + 1) = (Partial_Sums (P * ASeq)) . (k + 1)
then A34: k + 0 < ((len (P * FSeq)) - 1) + 1 by XREAL_1:8;
then A35: k < len FSeq by Th61;
k + 1 <= ((len (P * FSeq)) - 1) + 1 by A33, XREAL_1:7;
hence seq2 . (k + 1) = seq1 . (k + 1) by A30
.= (seq1 . k) + ((P * FSeq) . (k + 1)) by A26, A33, A34
.= ((Partial_Sums (P * ASeq)) . k) + ((P * FSeq) . (k + 1)) by A30, A32, A33, A34
.= (Partial_Sums (P * ASeq)) . (k + 1) by A8, A23, A33, A35, NAT_1:14 ;
:: thesis: verum
end;
case ( k <> 0 & not k <= (len (P * FSeq)) - 1 ) ; :: thesis: seq2 . (k + 1) = (Partial_Sums (P * ASeq)) . (k + 1)
then A36: k + 1 > ((len (P * FSeq)) - 1) + 1 by XREAL_1:8;
then k + 1 >= (len (P * FSeq)) + 1 by NAT_1:13;
then consider i being Nat such that
A37: k + 1 = ((len (P * FSeq)) + 1) + i by NAT_1:10;
thus seq2 . (k + 1) = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) by A30, A36
.= (Partial_Sums (P * ASeq)) . (len FSeq) by Th61
.= (Partial_Sums (P * ASeq)) . (((len FSeq) + 1) + i) by A18
.= (Partial_Sums (P * ASeq)) . (k + 1) by A37, Th61 ; :: thesis: verum
end;
end;
end;
hence S3[k + 1] ; :: thesis: verum
end;
seq2 . 0 = (P * ASeq) . 0 by A4, A30
.= (Partial_Sums (P * ASeq)) . 0 by SERIES_1:def 1 ;
then A38: S3[ 0 ] ;
A39: for k being Nat holds S3[k] from NAT_1:sch 2(A38, A31);
 len (P * FSeq) <> 0 by A23, Th61;
then seq2 . (len (P * FSeq)) = Sum (P * FSeq) by A27, A30;
hence Sum (P * FSeq) = (Partial_Sums (P * ASeq)) . (len (P * FSeq)) by A39
.= Sum (P * ASeq) by A21, Th61 ;
:: thesis: verum
end;
end;
end;
hence Sum (P * FSeq) = Sum (P * ASeq) ; :: thesis: verum
end;
Partial_Sums (P * ASeq) is convergent by A19, Th3;
then lim (Partial_Sums (P * (Partial_Diff_Union ASeq))) <= lim (Partial_Sums (P * ASeq)) by A16, A15, SEQ_2:18;
then Sum (P * (Partial_Diff_Union ASeq)) <= lim (Partial_Sums (P * ASeq)) by SERIES_1:def 3;
then Sum (P * (Partial_Diff_Union ASeq)) <= Sum (P * ASeq) by SERIES_1:def 3;
then P . (Union (Partial_Diff_Union ASeq)) <= Sum (P * ASeq) by Th46;
hence ( Partial_Sums (P * ASeq) is convergent & Sum (P * ASeq) = (Partial_Sums (P * ASeq)) . (len FSeq) & P . (Union ASeq) <= Sum (P * ASeq) & Sum (P * FSeq) = Sum (P * ASeq) ) by A19, A20, A22, Th3, Th20, SERIES_1:def 3; :: thesis: verum