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 = inf (proj1 .: C1) by Th48;
A3: ( not proj1 .: C2 is empty & proj1 .: C2 is bounded_below ) by Th46;
A4: ( not proj1 .: C1 is empty & proj1 .: C1 is bounded_below ) by Th46;
A5: W-bound C2 = inf (proj1 .: C2) by Th48;
thus W-bound S = inf (proj1 .: S) by Th48
.= inf ((proj1 .: C1) \/ (proj1 .: C2)) by A1, RELAT_1:153
.= min (W-bound C1),(W-bound C2) by A2, A5, A4, A3, Th52 ; :: thesis: verum