theorem Th10:
  for G being finite Group, a being Element of G holds card gr {a
  |^s} = card gr {a|^k} & a|^k in gr {a|^s} implies gr {a|^s} = gr {a|^k}
proof
  let G be finite Group, a be Element of G;
  assume
A1: card gr {a|^s} = card gr {a|^k};
  assume a|^k in gr {a|^s};
  then reconsider h=a|^k as Element of gr {a|^s} by STRUCT_0:def 5;
  card gr {h} = card gr {a|^s} by A1,Th3;
  hence gr {a|^s} = gr {h} by GROUP_2:73
    .= gr {a|^k} by Th3;
end;
