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)

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

A _\/_ (B _\/_ C) c= (A _\/_ B) _\/_ C
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 UNION (A,B) & Y in C & x = X \/ Y ) by SETFAM_1:def 4;

consider Z, W being set such that

A3: ( Z in A & W in B & X = Z \/ W ) by A2, SETFAM_1:def 4;

W \/ Y in UNION (B,C) by A2, A3, SETFAM_1:def 4;

then Z \/ (W \/ Y) in UNION (A,(UNION (B,C))) by A3, SETFAM_1:def 4;

hence x in A _\/_ (B _\/_ C) by A2, A3, XBOOLE_1:4; :: thesis: verum

end;assume x in (A _\/_ B) _\/_ C ; :: thesis: x in A _\/_ (B _\/_ C)

then consider X, Y being set such that

A2: ( X in UNION (A,B) & Y in C & x = X \/ Y ) by SETFAM_1:def 4;

consider Z, W being set such that

A3: ( Z in A & W in B & X = Z \/ W ) by A2, SETFAM_1:def 4;

W \/ Y in UNION (B,C) by A2, A3, SETFAM_1:def 4;

then Z \/ (W \/ Y) in UNION (A,(UNION (B,C))) by A3, SETFAM_1:def 4;

hence x in A _\/_ (B _\/_ C) by A2, A3, XBOOLE_1:4; :: thesis: verum

proof

hence
(A _\/_ B) _\/_ C = A _\/_ (B _\/_ C)
by A1; :: thesis: verum
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 UNION (B,C) & x = X \/ Y ) by SETFAM_1:def 4;

consider Z, W being set such that

A5: ( Z in B & W in C & Y = Z \/ W ) by A4, SETFAM_1:def 4;

X \/ Z in UNION (A,B) by A4, A5, SETFAM_1:def 4;

then (X \/ Z) \/ W in UNION ((UNION (A,B)),C) by A5, SETFAM_1:def 4;

hence x in (A _\/_ B) _\/_ C by A4, A5, XBOOLE_1:4; :: thesis: verum

end;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 UNION (B,C) & x = X \/ Y ) by SETFAM_1:def 4;

consider Z, W being set such that

A5: ( Z in B & W in C & Y = Z \/ W ) by A4, SETFAM_1:def 4;

X \/ Z in UNION (A,B) by A4, A5, SETFAM_1:def 4;

then (X \/ Z) \/ W in UNION ((UNION (A,B)),C) by A5, SETFAM_1:def 4;

hence x in (A _\/_ B) _\/_ C by A4, A5, XBOOLE_1:4; :: thesis: verum