let G be non empty multMagma ; :: thesis: 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 ) )
let A be set ; :: thesis: ( ( 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 ) )
A1: ( H₁( .: (G,A)) = (H₁(G),H₃(G)) .: A & H₃( .: (G,A)) = Funcs (A,H₃(G)) ) by Th17;
thus ( G is commutative implies .: (G,A) is commutative ) by A1, Th7; :: thesis: ( ( 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 ) )
thus ( G is associative implies .: (G,A) is associative ) by A1, Th8; :: thesis: ( ( 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 ) )
thus ( G is idempotent implies .: (G,A) is idempotent ) by A1, Th11; :: thesis: ( ( 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 ) )
thus ( G is invertible implies .: (G,A) is invertible ) by A1, Th12; :: thesis: ( ( G is cancelable implies .: (G,A) is cancelable ) & ( G is uniquely-decomposable implies .: (G,A) is uniquely-decomposable ) )
thus ( G is cancelable implies .: (G,A) is cancelable ) by A1, Th13; :: thesis: ( G is uniquely-decomposable implies .: (G,A) is uniquely-decomposable )
assume H₁(G) is uniquely-decomposable ; :: according to MONOID_0:def 20 :: thesis: .: (G,A) is uniquely-decomposable
hence H₁( .: (G,A)) is uniquely-decomposable by A1, Th14; :: according to MONOID_0:def 20 :: thesis: verum