reserve X for set,
  Y for non empty set;

theorem Th4:
  for f being Function of X,Y, A being Subset of X st f is
  one-to-one holds f.:A` c= (f.:A)`
proof
  let f be Function of X,Y, A be Subset of X such that
A1: f is one-to-one;
  let e be object;
  assume
A2: e in f.:A`;
  then reconsider y = e as Element of Y;
  consider x1 being object such that
A3: x1 in X and
A4: x1 in A` and
A5: y = f.x1 by A2,FUNCT_2:64;
  assume not e in (f.:A)`;
  then
A6: ex x2 being object st x2 in X & x2 in A & y = f.x2
       by FUNCT_2:64,SUBSET_1:29;
  not x1 in A by A4,XBOOLE_0:def 5;
  hence contradiction by A1,A3,A5,A6,FUNCT_2:19;
end;
