defpred S₁[ Nat, Element of REAL ] means ex i being Element of NAT st
( $1 = i & $2 = integral ((proj i,n) * f),a,b );
PN: for i being Nat st i in Seg n holds
ex x being Element of REAL st S₁[i,x]

proof

let i be Nat; :: thesis: ( i in Seg n implies ex x being Element of REAL st S₁[i,x] )
assume i in Seg n ; :: thesis: ex x being Element of REAL st S₁[i,x]
then reconsider ii = i as Element of NAT ;
consider x being Element of REAL such that
P1: x = integral ((proj ii,n) * f),a,b ;
take x ; :: thesis: S₁[i,x]
thus S₁[i,x] by P1; :: thesis: verum

end;

consider p being FinSequence of REAL such that
A1: ( dom p = Seg n & ( for i being Nat st i in Seg n holds
S₁[i,p . i] ) ) from FINSEQ_1:sch 5(PN);
take p ; :: thesis: ( p is Element of REAL n & dom p = Seg n & ( for i being Element of NAT st i in Seg n holds
p . i = integral ((proj i,n) * f),a,b ) )
A2: len p = n by A1, FINSEQ_1:def 3;
for i being Element of NAT st i in Seg n holds
p . i = integral ((proj i,n) * f),a,b

proof

let i be Element of NAT ; :: thesis: ( i in Seg n implies p . i = integral ((proj i,n) * f),a,b )
assume i in Seg n ; :: thesis: p . i = integral ((proj i,n) * f),a,b
then S₁[i,p . i] by A1;
hence p . i = integral ((proj i,n) * f),a,b ; :: thesis: verum

end;

hence ( p is Element of REAL n & dom p = Seg n & ( for i being Element of NAT st i in Seg n holds
p . i = integral ((proj i,n) * f),a,b ) ) by A1, A2, FINSEQ_2:110; :: thesis: verum