let G be non empty multMagma ; :: thesis: for A, B being Subset of G st G is commutative Group holds

A * B = B * A

let A, B be Subset of G; :: thesis: ( G is commutative Group implies A * B = B * A )

assume A1: G is commutative Group ; :: thesis: A * B = B * A

thus A * B c= B * A :: according to XBOOLE_0:def 10 :: thesis: B * A c= A * B

assume x in B * A ; :: thesis: x in A * B

then consider g, h being Element of G such that

A4: x = g * h and

A5: ( g in B & h in A ) ;

x = h * g by A1, A4, Lm2;

hence x in A * B by A5; :: thesis: verum

A * B = B * A

let A, B be Subset of G; :: thesis: ( G is commutative Group implies A * B = B * A )

assume A1: G is commutative Group ; :: thesis: A * B = B * A

thus A * B c= B * A :: according to XBOOLE_0:def 10 :: thesis: B * A c= A * B

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in B * A or x in A * B )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A * B or x in B * A )

assume x in A * B ; :: thesis: x in B * A

then consider g, h being Element of G such that

A2: x = g * h and

A3: ( g in A & h in B ) ;

x = h * g by A1, A2, Lm2;

hence x in B * A by A3; :: thesis: verum

end;assume x in A * B ; :: thesis: x in B * A

then consider g, h being Element of G such that

A2: x = g * h and

A3: ( g in A & h in B ) ;

x = h * g by A1, A2, Lm2;

hence x in B * A by A3; :: thesis: verum

assume x in B * A ; :: thesis: x in A * B

then consider g, h being Element of G such that

A4: x = g * h and

A5: ( g in B & h in A ) ;

x = h * g by A1, A4, Lm2;

hence x in A * B by A5; :: thesis: verum