let S be Subset of (TOP-REAL 2); :: thesis: for C1, C2 being non empty compact Subset of (TOP-REAL 2) st S = C1 \/ C2 holds
N-bound S = max (N-bound C1),(N-bound C2)
let C1, C2 be non empty compact Subset of (TOP-REAL 2); :: thesis: ( S = C1 \/ C2 implies N-bound S = max (N-bound C1),(N-bound C2) )
assume A1: S = C1 \/ C2 ; :: thesis: N-bound S = max (N-bound C1),(N-bound C2)
A2: N-bound C1 = upper_bound (proj2 .: C1) by Th50;
A3: ( not proj2 .: C2 is empty & proj2 .: C2 is bounded_above ) ;
A4: ( not proj2 .: C1 is empty & proj2 .: C1 is bounded_above ) ;
A5: N-bound C2 = upper_bound (proj2 .: C2) by Th50;
thus N-bound S = upper_bound (proj2 .: S) by Th50
.= upper_bound ((proj2 .: C1) \/ (proj2 .: C2)) by A1, RELAT_1:153
.= max (N-bound C1),(N-bound C2) by A2, A5, A4, A3, Th53 ; :: thesis: verum