reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;

theorem Th20:
  for G being finite Group
  for H1,H2 being strict Subgroup of G
  st H1,H2 are_isomorphic
  holds index H1 = index H2
proof
  let G be finite Group;
  let H1,H2 be strict Subgroup of G;
  assume A1: H1,H2 are_isomorphic;
  card H1 * index H1 = card G by GROUP_2:147
                    .= card H2 * index H2 by GROUP_2:147;
  then index H1 * card H1 = index H2 * card H1 by A1,GROUP_6:73;
  hence index H1 = index H2 by XCMPLX_1:5;
end;
