defpred S₁[ Nat, Element of REAL ] means ex i being Element of NAT st
( $1 = i & $2 = integral ((proj (i,n)) * f) );
A1: 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
A2: x = integral ((proj (ii,n)) * f) ;
take x ; :: thesis: S₁[i,x]
thus S₁[i,x] by A2; :: thesis: verum

end;

consider p being FinSequence of REAL such that
A3: ( dom p = Seg n & ( for i being Nat st i in Seg n holds
S₁[i,p . i] ) ) from FINSEQ_1:sch 5(A1);
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) ) )
A4: for i being Element of NAT st i in Seg n holds
p . i = integral ((proj (i,n)) * f)

proof

let i be Element of NAT ; :: thesis: ( i in Seg n implies p . i = integral ((proj (i,n)) * f) )
assume i in Seg n ; :: thesis: p . i = integral ((proj (i,n)) * f)
then S₁[i,p . i] by A3;
hence p . i = integral ((proj (i,n)) * f) ; :: thesis: verum

end;

len p = n by A3, FINSEQ_1:def 3;
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) ) ) by A3, A4, FINSEQ_2:110; :: thesis: verum