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
consider A', B' being Subset of (TopSpaceMetr (Euclid n)) such that
A2: ( A = A' & B = B' ) and
A3: dist_min A,B = min_dist_min A',B' by Def1;
TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
then ( A' is compact & B' is compact ) by A2, COMPTS_1:33;
hence dist_min A,B = 0 by A1, A2, A3, Th12; :: thesis: verum