let n be Element of NAT ; :: thesis: for P, Q being non empty Subset of (TOP-REAL n) st P is compact & Q is compact & HausDist P,Q = 0 holds
P = Q
let P, Q be non empty Subset of (TOP-REAL n); :: thesis: ( P is compact & Q is compact & HausDist P,Q = 0 implies P = Q )
assume that
A1: ( P is compact & Q is compact ) and
A2: HausDist P,Q = 0 ; :: thesis: P = Q
A3: 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)) ;
A4: HausDist P1,Q1 = 0 by A2, Def3;
( P1 is compact & Q1 is compact ) by A1, A3, COMPTS_1:33;
hence P = Q by A4, Th39; :: thesis: verum