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 Th18;
thus ( G is commutative implies .: (G,A) is commutative ) :: 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 ) )

proof

assume H₁(G) is commutative ; :: according to MONOID_0:def 11 :: thesis: .: (G,A) is commutative
hence H₁( .: (G,A)) is commutative by A1, Th8; :: according to MONOID_0:def 11 :: thesis: verum

end;

thus ( G is associative implies .: (G,A) is associative ) :: 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 ) )

proof

assume H₁(G) is associative ; :: according to MONOID_0:def 12 :: thesis: .: (G,A) is associative
hence H₁( .: (G,A)) is associative by A1, Th9; :: according to MONOID_0:def 12 :: thesis: verum

end;

thus ( G is idempotent implies .: (G,A) is idempotent ) :: 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 ) )

proof

assume H₁(G) is idempotent ; :: according to MONOID_0:def 13 :: thesis: .: (G,A) is idempotent
hence H₁( .: (G,A)) is idempotent by A1, Th12; :: according to MONOID_0:def 13 :: thesis: verum

end;

thus ( G is invertible implies .: (G,A) is invertible ) :: thesis: ( ( G is cancelable implies .: (G,A) is cancelable ) & ( G is uniquely-decomposable implies .: (G,A) is uniquely-decomposable ) )

proof

assume H₁(G) is invertible ; :: according to MONOID_0:def 16 :: thesis: .: (G,A) is invertible
hence H₁( .: (G,A)) is invertible by A1, Th13; :: according to MONOID_0:def 16 :: thesis: verum

end;

thus ( G is cancelable implies .: (G,A) is cancelable ) :: thesis: ( G is uniquely-decomposable implies .: (G,A) is uniquely-decomposable )

proof

assume H₁(G) is cancelable ; :: according to MONOID_0:def 19 :: thesis: .: (G,A) is cancelable
hence H₁( .: (G,A)) is cancelable by A1, Th14; :: according to MONOID_0:def 19 :: thesis: verum

end;

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, Th15; :: according to MONOID_0:def 20 :: thesis: verum