let n be Element of NAT ; :: thesis: for A, B being non empty compact Subset of (TOP-REAL n) st A misses B holds
dist_min A,B > 0
let A, B be non empty compact Subset of (TOP-REAL n); :: thesis: ( A misses B implies dist_min A,B > 0 )
assume A1: A misses B ; :: thesis: dist_min A,B > 0
A2: 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
A3: A = A' and
A4: B = B' and
A5: dist_min A,B = min_dist_min A',B' by JORDAN1K:def 1;
A6: A' is compact by A2, A3, COMPTS_1:33;
B' is compact by A2, A4, COMPTS_1:33;
hence dist_min A,B > 0 by A1, A3, A4, A5, A6, JGRAPH_1:55; :: thesis: verum