defpred S1[ Nat, Real] means ex r being Element of REAL st
( r in rng (f | (divset (D,$1))) & $2 = r * (vol (divset (D,$1))) );
A1:
Seg (len D) = dom D
by FINSEQ_1:def 3;
A2:
for k being Nat st k in Seg (len D) holds
ex x being Element of REAL st S1[k,x]
proof
let k be
Nat;
( k in Seg (len D) implies ex x being Element of REAL st S1[k,x] )
assume
k in Seg (len D)
;
ex x being Element of REAL st S1[k,x]
then A3:
k in dom D
by FINSEQ_1:def 3;
dom f = A
by FUNCT_2:def 1;
then
dom (f | (divset (D,k))) = divset (
D,
k)
by A3, INTEGRA1:8, RELAT_1:62;
then
not
rng (f | (divset (D,k))) is
empty
by RELAT_1:42;
then consider r being
object such that A4:
r in rng (f | (divset (D,k)))
;
reconsider r =
r as
Element of
REAL by A4;
r * (vol (divset (D,k))) is
Element of
REAL
by XREAL_0:def 1;
hence
ex
x being
Element of
REAL st
S1[
k,
x]
by A4;
verum
end;
consider p being FinSequence of REAL such that
A5:
( dom p = Seg (len D) & ( for k being Nat st k in Seg (len D) holds
S1[k,p . k] ) )
from FINSEQ_1:sch 5(A2);
len p = len D
by A5, FINSEQ_1:def 3;
hence
ex b1 being FinSequence of REAL st
( len b1 = len D & ( for i being Nat st i in dom D holds
ex r being Element of REAL st
( r in rng (f | (divset (D,i))) & b1 . i = r * (vol (divset (D,i))) ) ) )
by A5, A1; verum