reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;
reserve G for non empty multMagma;

theorem Th23:
  for G being non empty multMagma, A being set holds (G is
commutative implies .:(G,A) is commutative) & (G is associative implies .:(G,A)
  is associative) & (G is idempotent implies .:(G,A) is idempotent) & (G is
invertible implies .:(G,A) is invertible) & (G is cancelable implies .:(G,A) is
  cancelable) & (G is uniquely-decomposable implies .:(G,A) is
  uniquely-decomposable)
proof
  let G;
  let A be set;
A1: op(.:(G,A)) = (op(G), carr(G)).:A & carr(.:(G,A)) = Funcs(A, carr(G)) by
Th17;
  thus G is commutative implies .:(G,A) is commutative
  by A1,Th7;
  thus G is associative implies .:(G,A) is associative
  by A1,Th8;
  thus G is idempotent implies .:(G,A) is idempotent
  by A1,Th11;
  thus G is invertible implies .:(G,A) is invertible
  by A1,Th12;
  thus G is cancelable implies .:(G,A) is cancelable
  by A1,Th13;
  assume op(G) is uniquely-decomposable;
  hence op(.:(G,A)) is uniquely-decomposable by A1,Th14;
end;
