let n be Element of NAT ; for P, Q being non empty Subset of (TOP-REAL n) st P is compact & Q is compact holds
HausDist (P,Q) >= 0
let P, Q be non empty Subset of (TOP-REAL n); ( P is compact & Q is compact implies HausDist (P,Q) >= 0 )
A1:
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n)
by EUCLID:def 8;
then reconsider P1 = P, Q1 = Q as non empty Subset of (TopSpaceMetr (Euclid n)) ;
assume
( P is compact & Q is compact )
; HausDist (P,Q) >= 0
then
( P1 is compact & Q1 is compact )
by A1, COMPTS_1:23;
then
HausDist (P1,Q1) >= 0
by Th35;
hence
HausDist (P,Q) >= 0
by Def3; verum