
theorem Th13:
for A be non empty closed_interval Subset of REAL,
f be Function of A,COMPLEX holds
    f is bounded iff (Re f is bounded & Im f is bounded)
proof
let A be non empty closed_interval Subset of REAL,
    f be Function of A,COMPLEX;
dom f = A by FUNCT_2:def 1; then
reconsider f0=f as PartFunc of REAL,COMPLEX by RELSET_1:5;
f0 is bounded iff Re f0 is bounded & Im f0 is bounded by Th11;
hence thesis;
end;
