 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;
reserve M for non empty multLoopStr;

theorem Th19:
  for G being non empty multMagma, M being MonoidalExtension of G holds
    (G is unital implies M is unital) &
    (G is commutative implies M is commutative) &
    (G is associative implies M is associative) &
    (G is invertible implies M is invertible) &
    (G is uniquely-decomposable implies M is uniquely-decomposable) &
    (G is cancelable implies M is cancelable)
proof
  let G be non empty multMagma, M be MonoidalExtension of G;
A1: the multMagma of M = the multMagma of G by Def22;
  thus G is unital implies M is unital by A1;
  thus G is commutative implies M is commutative by A1;
  thus G is associative implies M is associative by A1;
  thus G is invertible implies M is invertible by A1;
  thus G is uniquely-decomposable implies M is uniquely-decomposable
  by A1;
  assume op(G) is cancelable;
  hence op(M) is cancelable by A1;
end;
