
theorem
for f being PartFunc of REAL,COMPLEX,
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 f be PartFunc of REAL,COMPLEX,
    A be non empty closed_interval Subset of REAL, a,b be Real;
assume A1: A=[.b,a.];
A2: Re (integral(f,a,b)) = integral((Re f), a,b)
  & Im (integral(f,a,b)) = integral((Im f), a,b) by COMPLEX1:12;
A3: Re (-integral(f,A)) = -(Re (integral(f,A))) by COMPLEX1:17
                       .= -integral((Re f),A) by COMPLEX1:12;
A4: Im (-integral(f,A)) = -(Im (integral(f,A))) by COMPLEX1:17
                       .=-integral((Im f),A) by COMPLEX1:12;
A5: Re (-integral(f,A)) = Re (integral(f,a,b)) by A3,A1,A2,INTEGRA5:20;
Im (-integral(f,A)) = Im (integral(f,a,b)) by A4,A1,A2,INTEGRA5:20;
hence thesis by A5;
end;
