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;
reserve E for set,
  A for Action of O,E,
  C for Subset of G,
  N1 for normal StableSubgroup of H1;

theorem Th75:
  the carrier of H1 = the carrier of H2 implies the HGrWOpStr of
  H1 = the HGrWOpStr of H2
proof
  reconsider H19=H1,H29=H2 as Subgroup of G by Def7;
A1: dom the action of H2 = O by FUNCT_2:def 1
    .= dom the action of H1 by FUNCT_2:def 1;
  assume
A2: the carrier of H1 = the carrier of H2;
A3: now
    let x be object;
    assume
A4: x in dom the action of H2;
    then reconsider o=x as Element of O;
A5: H1^o = (the action of H1).o by A4,Def6;
    H1^o = (G^o)|the carrier of H2 by A2,Def7
      .= H2^o by Def7;
    hence (the action of H1).x = (the action of H2).x by A4,A5,Def6;
  end;
  the multMagma of H19 = the multMagma of H29 by A2,GROUP_2:59;
  hence thesis by A1,A3,FUNCT_1:2;
end;
