reserve i for Element of NAT;

theorem Th14:
  for G,H being strict Group, h being Homomorphism of G,H for A,B
  being Subset of G holds h.:A* h.:B=h.:(A*B)
proof
  let G,H be strict Group, h be Homomorphism of G,H;
  let A,B be Subset of G;
  now
    let z be object;
    assume z in h.:(A)*h.:(B);
    then consider z1,z2 being Element of H such that
A1: z=z1 *z2 and
A2: z1 in h.:(A) and
A3: z2 in h.:(B);
    consider z4 being object such that
A4: z4 in the carrier of G and
A5: z4 in B and
A6: z2=h.z4 by A3,FUNCT_2:64;
    reconsider z4 as Element of G by A4;
    consider z3 being object such that
A7: z3 in the carrier of G and
A8: z3 in A and
A9: z1=h.z3 by A2,FUNCT_2:64;
    reconsider z3 as Element of G by A7;
A10: z3 * z4 in (A) *(B) by A8,A5;
    z=h.(z3* z4) by A1,A9,A6,GROUP_6:def 6;
    hence z in h.:((A) *(B)) by A10,FUNCT_2:35;
  end;
  then
A11: h.:(A)*h.:(B)c= h.:((A) *(B));
  now
    let z be object;
    assume z in h.:((A) *(B));
    then consider x being object such that
A12: x in the carrier of G and
A13: x in (A)*(B) and
A14: z=h.x by FUNCT_2:64;
    reconsider x as Element of G by A12;
    consider z1,z2 being Element of G such that
A15: x=z1 *z2 and
A16: z1 in(A) & z2 in (B) by A13;
A17: h.z1 in h.:(A) & h.z2 in h.:(B) by A16,FUNCT_2:35;
    z=h.z1 * h.z2 by A14,A15,GROUP_6:def 6;
    hence z in h.:(A)* h.:(B) by A17;
  end;
  then h.:((A) *(B)) c=h.:(A)*h.:(B);
  hence thesis by A11,XBOOLE_0:def 10;
end;
