defpred S_{1}[ 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 S_{1}[k,x]

A5: ( dom p = Seg (len D) & ( for k being Nat st k in Seg (len D) holds

S_{1}[k,p . k] ) )
from FINSEQ_1:sch 5(A2);

len p = len D by A5, FINSEQ_1:def 3;

hence ex b_{1} being FinSequence of REAL st

( len b_{1} = 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))) & b_{1} . i = r * (vol (divset (D,i))) ) ) )
by A5, A1; :: thesis: verum

( 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 S

proof

consider p being FinSequence of REAL such that
let k be Nat; :: thesis: ( k in Seg (len D) implies ex x being Element of REAL st S_{1}[k,x] )

assume k in Seg (len D) ; :: thesis: ex x being Element of REAL st S_{1}[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 S_{1}[k,x]
by A4; :: thesis: verum

end;assume k in Seg (len D) ; :: thesis: ex x being Element of REAL st S

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 S

A5: ( dom p = Seg (len D) & ( for k being Nat st k in Seg (len D) holds

S

len p = len D by A5, FINSEQ_1:def 3;

hence ex b

( len b

ex r being Element of REAL st

( r in rng (f | (divset (D,i))) & b