reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

theorem Th23:
  for G,H being strict Group, h being Homomorphism of G,H for A
  being strict Subgroup of G for a,b being Element of G
  holds h.a * h.b * h.:A = h.:(a * b * A) & h.:A * h.a * h.b = h.:(A * a * b)
proof
  let G,H be strict Group;
  let h be Homomorphism of G,H;
  let A be strict Subgroup of G;
  let a,b be Element of G;
  thus h.a * h.b * h.:A = h. (a * b) * h.:A by GROUP_6:def 6
                    .= h.:(a * b * A) by GRSOLV_1:13;
  thus h.:A * h.a * h.b = h.:A * (h.a * h.b) by GROUP_2:107
                  .= h.:A * h. (a * b) by GROUP_6:def 6
                  .= h.:(A * (a * b)) by GRSOLV_1:13
                  .= h.:(A * a * b) by GROUP_2:107;
end;
