theorem Th73:
  G is Abelian addGroup implies for H being strict Subgroup of G
  for a holds H * a = H
proof
  assume
A1: G is Abelian addGroup;
  let H be strict Subgroup of G;
  let a;
  the carrier of H * a = (-a) + H + a by ThB59
    .= H + (-a) + a by A1,Th112
    .= H + ((-a) + a) by ThA107
    .= H + 0_G by Def5
    .= the carrier of H by ThB109;
  hence thesis by Th59;
end;
