reserve a,x,y for object, A,B for set,
  l,m,n for Nat;

theorem Th4:
  for X,Y be non empty set, f being Function of X,Y st f is
  one-to-one for x being Element of X, A being Subset of X, B being Subset of Y
  st f.x in f.:A \ B holds x in A \ f"B
proof
  let X,Y be non empty set, f be Function of X,Y such that
A1: f is one-to-one;
  let x be Element of X, A be Subset of X, B be Subset of Y;
  assume
A2: f.x in f.:A \ B;
A3: now
    assume x in f"B;
    then f.x in B by FUNCT_1:def 7;
    hence contradiction by A2,XBOOLE_0:def 5;
  end;
  f.x in f.:A by A2,XBOOLE_0:def 5;
  then x in A by A1,Th3;
  hence thesis by A3,XBOOLE_0:def 5;
end;
