for G being non empty multMagma
for 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 ) )