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
W-bound S = min ((W-bound C1),(W-bound C2))
let C1, C2 be non empty compact Subset of (TOP-REAL 2); :: thesis: ( S = C1 \/ C2 implies W-bound S = min ((W-bound C1),(W-bound C2)) )
assume A1: S = C1 \/ C2 ; :: thesis: W-bound S = min ((W-bound C1),(W-bound C2))
A2: W-bound C1 = lower_bound (proj1 .: C1) by Th48;
A3: ( not proj1 .: C2 is empty & proj1 .: C2 is bounded_below ) ;
A5: W-bound C2 = lower_bound (proj1 .: C2) by Th48;
thus W-bound S = lower_bound (proj1 .: S) by Th48
.= lower_bound ((proj1 .: C1) \/ (proj1 .: C2)) by A1, RELAT_1:120
.= min ((W-bound C1),(W-bound C2)) by A2, A5, A3, SEQ_4:142 ; :: thesis: verum