 reserve U for set,
         X, Y for Subset of U;
 reserve U for non empty set,
         A, B, C for non empty IntervalSet of U;

theorem Th24:
  A _/\_ B _/\_ C = A _/\_ (B _/\_ C)
  proof
A1: A _/\_ B _/\_ C c= A _/\_ (B _/\_ C)
    proof
      let x be object;
      assume x in A _/\_ B _/\_ C; then
      consider X, Y being set such that
A2:   X in INTERSECTION (A, B) & Y in C & x = X /\ Y by SETFAM_1:def 5;
      consider Z, W being set such that
A3:   Z in A & W in B & X = Z /\ W by A2,SETFAM_1:def 5;
      W /\ Y in INTERSECTION (B,C) by A2,A3,SETFAM_1:def 5; then
      Z /\ (W /\ Y) in INTERSECTION (A,INTERSECTION (B,C))
        by A3,SETFAM_1:def 5;
      hence thesis by A2,A3,XBOOLE_1:16;
    end;
    A _/\_ (B _/\_ C) c= A _/\_ B _/\_ C
    proof
      let x be object;
      assume x in A _/\_ (B _/\_ C); then
      consider X, Y being set such that
A4:   X in A & Y in INTERSECTION (B, C) & x = X /\ Y by SETFAM_1:def 5;
      consider Z, W being set such that
A5:   Z in B & W in C & Y = Z /\ W by A4,SETFAM_1:def 5;
      X /\ Z in INTERSECTION (A,B) by A4,A5,SETFAM_1:def 5; then
      (X /\ Z) /\ W in INTERSECTION (INTERSECTION (A,B), C)
        by A5,SETFAM_1:def 5;
      hence thesis by A4,A5,XBOOLE_1:16;
    end;
    hence thesis by A1;
  end;
