reserve x,O for set,
  o for Element of O,
  G,H,I for GroupWithOperators of O,
  A, B for Subset of G,
  N for normal StableSubgroup of G,
  H1,H2,H3 for StableSubgroup of G,
  g1,g2 for Element of G,
  h1,h2 for Element of H1,
  h for Homomorphism of G,H;

theorem Th52:
  for H being strict GroupWithOperators of O, h being Homomorphism
of G,H st h is onto holds for c being Element of H ex a being Element of G st h
  .a = c
proof
  let H be strict GroupWithOperators of O;
  let h be Homomorphism of G,H;
  assume
A1: h is onto;
  let c be Element of H;
  rng h = the carrier of H by A1;
  then consider a be object such that
A2: a in dom h and
A3: c = h.a by FUNCT_1:def 3;
  reconsider a as Element of G by A2;
  take a;
  thus thesis by A3;
end;
