reserve i for Integer,
  a, b, r, s for Real;

theorem
  for f being Function, x, X being set st x in dom f & f.x in f.:X & f
  is one-to-one holds x in X
proof
  let f be Function, x, X be set;
  assume
A1: x in dom f;
  assume f.x in f.:X;
  then
A2: ex a being object st a in dom f & a in X & f.x = f.a by FUNCT_1:def 6;
  assume f is one-to-one;
  hence thesis by A1,A2,FUNCT_1:def 4;
end;
