theorem Th12:
  f is multiplicative implies f"Y is stable Subset of M
proof
  assume A1: f is multiplicative;
  for v,w being Element of M st v in f"Y & w in f"Y holds v*w in f"Y
  proof
    let v,w be Element of M;
    assume v in f"Y; then
    A2: v in dom f & f.v in Y by FUNCT_1:def 7;
    assume w in f"Y; then
    A3: w in dom f & f.w in Y by FUNCT_1:def 7;
    v*w in the carrier of M; then
    A4: v*w in dom f by FUNCT_2:def 1;
    f.v*f.w in Y by A2,A3,Def10; then
    f.(v*w) in Y by A1,GROUP_6:def 6;
    hence v*w in f"Y by A4,FUNCT_1:def 7;
  end;
  hence f"Y is stable Subset of M by Def10;
end;
