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 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 ;
W \/ Y in UNION (B,C) by ;
then Z \/ (W \/ Y) in UNION (A,(UNION (B,C))) by ;
hence x in A _\/_ (B _\/_ C) by ; :: 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 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 ;
X \/ Z in UNION (A,B) by ;
then (X \/ Z) \/ W in UNION ((UNION (A,B)),C) by ;
hence x in (A _\/_ B) _\/_ C by ; :: thesis: verum
end;
hence (A _\/_ B) _\/_ C = A _\/_ (B _\/_ C) by A1; :: thesis: verum