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 Th55:
  G,H are_isomorphic & H,I are_isomorphic implies G,I are_isomorphic
proof
  assume that
A1: G,H are_isomorphic and
A2: H,I are_isomorphic;
  consider g be Homomorphism of G,H such that
A3: g is bijective by A1;
  consider h1 be Homomorphism of H,I such that
A4: h1 is bijective by A2;
A5: rng h1 = the carrier of I by A4,FUNCT_2:def 3;
  rng g = the carrier of H by A3,FUNCT_2:def 3;
  then dom h1 = rng g by FUNCT_2:def 1;
  then rng(h1 * g) = the carrier of I by A5,RELAT_1:28;
  then h1 * g is onto;
  hence thesis by A3,A4;
end;
