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 Th3:
  (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_1:4
    .= 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_1:4
    .= a`1 \/ (b \/ c)`1
    .= (a \/ (b \/ c))`1;
  hence thesis by A1,DOMAIN_1:2;
end;
