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 Th58:
  for G,H being strict GroupWithOperators of O holds G,H
  are_isomorphic & G is trivial implies H is trivial
proof
  let G,H be strict GroupWithOperators of O;
  assume that
A1: G,H are_isomorphic and
A2: G is trivial;
  consider e be object such that
A3: the carrier of G = {e} by A2;
  consider g be Homomorphism of G,H such that
A4: g is bijective by A1;
  e in the carrier of G by A3,TARSKI:def 1;
  then
A5: e in dom g by FUNCT_2:def 1;
  the carrier of H = the carrier of Image g by A4,Th51
    .= Im(g,e) by A3,Def22
    .= {g.e} by A5,FUNCT_1:59;
  hence thesis;
end;
