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 st Y <> {} & ex g being Function of Y,X st
  g*f = id X holds f is one-to-one
proof
  let f be Function of X,Y;
  assume Y <> {};
  then
A1: dom f = X by Def1;
  given g being Function of Y,X such that
A2: g*f = id X;
  thus thesis by A2,A1,FUNCT_1:31;
end;
