reserve Z for set;

theorem
  for n be Element of NAT for f being PartFunc of REAL,REAL n, A being
  non empty closed_interval Subset of REAL, a,b be Real st A=[.a,b.]
   holds integral(f,A) = integral(f,a,b)
proof
  let n be Element of NAT;
  let f being PartFunc of REAL,REAL n,
  A being non empty closed_interval Subset of REAL,
  a,b be Real;
  assume
A1: A=[.a,b.];
A2: now
    let i be Nat;
    assume
A3: i in dom (integral(f,A));
    then reconsider k=i as Element of NAT;
    dom (integral(f,A)) = Seg n by Def17;
    then integral(f,A).k = integral((proj(k,n)*f),A) & integral(f,a,b).k =
    integral(( proj(k,n)*f), a,b) by A3,Def17,Def18;
    hence integral(f,A).i = integral(f,a,b).i by A1,INTEGRA5:19;
  end;
  dom (integral(f,A)) = Seg n by Def17
    .= dom (integral(f,a,b)) by Def18;
  hence integral(f,A)=integral(f,a,b) by A2,FINSEQ_1:13;
end;
