let n be Nat; :: thesis: for r being Real
for x being Point of (TOP-REAL n) holds Ball (x,r) c= cl_Ball (x,r)
let r be Real; :: thesis: for x being Point of (TOP-REAL n) holds Ball (x,r) c= cl_Ball (x,r)
let x be Point of (TOP-REAL n); :: thesis: Ball (x,r) c= cl_Ball (x,r)
reconsider e = x as Point of (Euclid n) by TOPREAL3:8;
( Ball (x,r) = Ball (e,r) & cl_Ball (x,r) = cl_Ball (e,r) ) by Th11, Th12;
hence Ball (x,r) c= cl_Ball (x,r) by METRIC_1:14; :: thesis: verum