theorem
  for G being non empty multMagma holds (G is commutative implies bool G
  is commutative) & (G is associative implies bool G is associative) & (G is
  uniquely-decomposable implies bool G is uniquely-decomposable)
proof
  let G be non empty multMagma;
A1: op(bool G) = op(G).:^2 & carr(bool G) = bool carr(G) by Th55;
  thus G is commutative implies bool G is commutative
  by A1,Th49;
  thus G is associative implies bool G is associative
  by A1,Th50;
  assume op(G) is uniquely-decomposable;
  hence op(bool G) is uniquely-decomposable by A1,Th54;
end;
