let n be Element of NAT ; :: thesis: for A, B being compact Subset of (TOP-REAL n) st A meets B holds
dist_min A,B = 0
let A, B be compact Subset of (TOP-REAL n); :: thesis: ( A meets B implies dist_min A,B = 0 )
assume A1: A meets B ; :: thesis: dist_min A,B = 0
X: TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
consider A', B' being Subset of (TopSpaceMetr (Euclid n)) such that
W1: ( A = A' & B = B' ) and
W2: dist_min A,B = min_dist_min A',B' by Def1;
( A' is compact & B' is compact ) by W1, X, COMPTS_1:33;
hence dist_min A,B = 0 by W2, A1, W1, Th12; :: thesis: verum