reserve i for Element of NAT;

theorem Th11:
  for G,H being Group, h being Homomorphism of G,H holds
  h.:((1).G)=(1).H & h.:((Omega).G)=(Omega).(Image h)
proof
  let G,H be Group;
  let h be Homomorphism of G,H;
A1: dom h=the carrier of G by FUNCT_2:def 1;
A2: the carrier of (1).H={1_H} by GROUP_2:def 7;
  the carrier of (1).G={1_G} by GROUP_2:def 7;
  then the carrier of h.:((1).G)=Im(h,1_G) by Th8
    .={h.(1_G)} by A1,FUNCT_1:59
    .= the carrier of (1).H by A2,GROUP_6:31;
  hence h.:((1).G)= (1).H by GROUP_2:59;
  the carrier of h.:((Omega).G)=h.:(the carrier of G) by Th8
    .=the carrier of (Omega).(Image h) by GROUP_6:def 10;
  hence thesis by GROUP_2:59;
end;
