theorem
  for G being finite addGroup, H being Subgroup of G holds card G = card H
  implies the addMagma of H = the addMagma of G
proof
  let G be finite addGroup, H be Subgroup of G;
  assume
A1: card G = card H;
  the carrier of H = the carrier of G
  proof
    assume the carrier of H <> the carrier of G;
    then the carrier of H c< the carrier of G by DefA5;
    hence thesis by A1,CARD_2:48;
  end;
  hence thesis by Th61;
end;
