theorem Th87:
  f"Y1 c= f"Y2 & Y1 c= rng f implies Y1 c= Y2
proof
  assume that
A1: f"Y1 c= f"Y2 and
A2: Y1 c= rng f;
  let y be object;
  assume
A3: y in Y1;
  then consider x being object such that
A4: x in dom f and
A5: y = f.x by A2,Def3;
  x in f"Y1 by A3,A4,A5,Def7;
  hence thesis by A1,A5,Def7;
end;
