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=[.b,a.]
 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=[.b,a.];
A2: now
    let i be Nat;
    assume
A3: i in dom (-integral(f,A));
    then reconsider k=i as Element of NAT;
A4: dom (integral(f,A))= Seg n by Def17;
A5: k in dom (integral(f,A)) by A3,VALUED_1:8;
    then
A6: integral(f,a,b).k = integral((proj(k,n)*f), a,b) by A4,Def18;
    (-integral(f,A)).k =-((integral(f,A)).k) by VALUED_1:8
      .=-integral((proj(k,n)*f),A) by A5,A4,Def17;
    hence (-integral(f,A)).i = integral(f,a,b).i by A1,A6,INTEGRA5:20;
  end;
  reconsider jj=1 as Real;
  dom (-integral(f,A)) = dom ((-jj)(#)integral(f,A)) by VALUED_1:def 6
    .= dom (integral(f,A)) by VALUED_1:def 5
    .= Seg n by Def17
    .= dom (integral(f,a,b)) by Def18;
  hence thesis by A2,FINSEQ_1:13;
end;
