reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem
  for f being Function of X,Y for g being Function of Y,X st X <> {} & Y
  <> {} & rng f = Y & g*f = id X & f is one-to-one holds g = f"
proof
  let f be Function of X,Y;
  let g be Function of Y,X;
  assume X <> {} & Y <> {};
  then dom f = X & dom g = Y by Def1;
  hence thesis by FUNCT_1:41;
end;
