defpred S₁[ Nat, set ] means ex c being Point of X st
( c in rng (f | (divset (D,$1))) & $2 = (vol (divset (D,$1))) * c );
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 the carrier of X st S₁[k,x]

proof

let k be Nat; :: thesis: ( k in Seg (len D) implies ex x being Element of the carrier of X st S₁[k,x] )
assume k in Seg (len D) ; :: thesis: ex x being Element of the carrier of X 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 c being object such that
A4: c in rng (f | (divset (D,k))) ;
reconsider c = c as Point of X by A4;
(vol (divset (D,k))) * c is Element of the carrier of X ;
hence ex x being Element of the carrier of X st S₁[k,x] by A4; :: thesis: verum

end;

consider p being FinSequence of the carrier of X 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 X st
( len b₁ = len D & ( for i being Nat st i in dom D holds
ex c being Point of X st
( c in rng (f | (divset (D,i))) & b₁ . i = (vol (divset (D,i))) * c ) ) ) by A5, A1; :: thesis: verum