let U be non empty set ; :: thesis: for A, B, C being non empty IntervalSet of U holds (A _/\_ B) _/\_ C = A _/\_ (B _/\_ C)
let A, B, C be non empty IntervalSet of U; :: thesis: (A _/\_ B) _/\_ C = A _/\_ (B _/\_ C)
A1: (A _/\_ B) _/\_ C c= A _/\_ (B _/\_ C)

proof

let x be object ; :: 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 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 x in A _/\_ (B _/\_ C) by A2, A3, XBOOLE_1:16; :: thesis: verum

end;

A _/\_ (B _/\_ C) c= (A _/\_ B) _/\_ C

proof

let x be object ; :: 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 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 x in (A _/\_ B) _/\_ C by A4, A5, XBOOLE_1:16; :: thesis: verum

end;

hence (A _/\_ B) _/\_ C = A _/\_ (B _/\_ C) by A1; :: thesis: verum