
theorem
for f being PartFunc of REAL,COMPLEX,
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 f be PartFunc of REAL,COMPLEX,
    A be non empty closed_interval Subset of REAL, a,b be Real;
assume A1: A=[.a,b.];
  Re (integral(f,A)) = integral((Re f),A)
& Im (integral(f,A)) = integral((Im f),A)
& Re (integral(f,a,b)) =integral((Re f), a,b)
& Im (integral(f,a,b)) =integral((Im f), a,b) by COMPLEX1:12; then
  Re (integral(f,A)) = Re (integral(f,a,b))
& Im (integral(f,A)) = Im (integral(f,a,b)) by A1,INTEGRA5:19;
hence integral(f,A)=integral(f,a,b);
end;
