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 Th45;
A3: N-bound C2 = upper_bound (proj2 .: C2) by Th45;
thus N-bound S = upper_bound (proj2 .: S) by Th45
.= upper_bound ((proj2 .: C1) \/ (proj2 .: C2)) by A1, RELAT_1:120
.= max ((N-bound C1),(N-bound C2)) by A2, A3, SEQ_4:143 ; :: thesis: verum