let X, Y, Z be set ; :: thesis: (X /\ Y) /\ Z = X /\ (Y /\ Z)
thus (X /\ Y) /\ Z c= X /\ (Y /\ Z) :: according to XBOOLE_0:def 10 :: thesis: X /\ (Y /\ Z) c= (X /\ Y) /\ Z
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (X /\ Y) /\ Z or x in X /\ (Y /\ Z) )
assume A1: x in (X /\ Y) /\ Z ; :: thesis: x in X /\ (Y /\ Z)
then A2: x in Z by XBOOLE_0:def 4;
A3: x in X /\ Y by A1, XBOOLE_0:def 4;
then A4: x in X by XBOOLE_0:def 4;
x in Y by A3, XBOOLE_0:def 4;
then x in Y /\ Z by A2, XBOOLE_0:def 4;
hence x in X /\ (Y /\ Z) by A4, XBOOLE_0:def 4; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X /\ (Y /\ Z) or x in (X /\ Y) /\ Z )
assume A5: x in X /\ (Y /\ Z) ; :: thesis: x in (X /\ Y) /\ Z
then A6: x in Y /\ Z by XBOOLE_0:def 4;
then A7: x in Y by XBOOLE_0:def 4;
A8: x in Z by A6, XBOOLE_0:def 4;
x in X by A5, XBOOLE_0:def 4;
then x in X /\ Y by A7, XBOOLE_0:def 4;
hence x in (X /\ Y) /\ Z by A8, XBOOLE_0:def 4; :: thesis: verum