theorem :: JORDAN1C:4

for C1 being non empty compact Subset of (TOP-REAL 2)
for C2, S being non empty Subset of (TOP-REAL 2) st S = C1 \/ C2 & not proj1 .: C2 is empty & proj1 .: C2 is bounded_below holds
W-bound S = min ((W-bound C1),(W-bound C2))