let G be non empty multMagma ; :: thesis: for A, B, C being Subset of G st G is associative holds
(A * B) * C = A * (B * C)
let A, B, C be Subset of G; :: thesis: ( G is associative implies (A * B) * C = A * (B * C) )
assume A1: G is associative ; :: thesis: (A * B) * C = A * (B * C)
thus (A * B) * C c= A * (B * C) :: according to XBOOLE_0:def 10 :: thesis: A * (B * C) c= (A * B) * C

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (A * B) * C or x in A * (B * C) )
assume x in (A * B) * C ; :: thesis: x in A * (B * C)
then consider g, h being Element of G such that
A2: x = g * h and
A3: g in A * B and
A4: h in C ;
consider g1, g2 being Element of G such that
A5: g = g1 * g2 and
A6: g1 in A and
A7: g2 in B by A3;
( x = g1 * (g2 * h) & g2 * h in B * C ) by A1, A2, A4, A5, A7, GROUP_1:def 4;
hence x in A * (B * C) by A6; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in A * (B * C) or x in (A * B) * C )
assume x in A * (B * C) ; :: thesis: x in (A * B) * C
then consider g, h being Element of G such that
A8: x = g * h and
A9: g in A and
A10: h in B * C ;
consider g1, g2 being Element of G such that
A11: h = g1 * g2 and
A12: g1 in B and
A13: g2 in C by A10;
( x = (g * g1) * g2 & g * g1 in A * B ) by A1, A8, A9, A11, A12, GROUP_1:def 4;
hence x in (A * B) * C by A13; :: thesis: verum