reserve A, B for non empty preBoolean set,
  x, y for Element of [:A,B:];
reserve X for set,
  a,b,c for Element of [:A,B:];

theorem
  (a /\ b) /\ c = a /\ (b /\ c)
proof
A1: ((a /\ b) /\ c)`2 = (a /\ b)`2 /\ c`2
    .= a`2 /\ b`2 /\ c`2
    .= a`2 /\ (b`2 /\ c`2) by XBOOLE_1:16
    .= a`2 /\ (b /\ c)`2
    .= (a /\ (b /\ c))`2;
  ((a /\ b) /\ c)`1 = (a /\ b)`1 /\ c`1
    .= a`1 /\ b`1 /\ c`1
    .= a`1 /\ (b`1 /\ c`1) by XBOOLE_1:16
    .= a`1 /\ (b /\ c)`1
    .= (a /\ (b /\ c))`1;
  hence thesis by A1,DOMAIN_1:2;
end;
