
theorem Th5:
  for M being non empty MetrSpace, P being Subset of TopSpaceMetr M
  , x being set st x in P holds (dist_min P) . x = 0
proof
  let M be non empty MetrSpace, P be Subset of TopSpaceMetr M, x be set;
  assume
A1: x in P;
  then reconsider x as Point of M by TOPMETR:12;
  set X = (dist x) .: P;
  reconsider X as non empty Subset of REAL by A1,TOPMETR:17;
  lower_bound ((dist x) .: P) = lower_bound [#] ((dist x) .: P) by
WEIERSTR:def 3
    .= lower_bound X by WEIERSTR:def 1;
  then
A2: (dist_min P) . x = lower_bound X by WEIERSTR:def 6;
A3: for p being Real st p in X holds p >= 0 by Th4;
  for q being Real st for p being Real st p in X holds p >=
  q holds 0 >= q by A1,Th3;
  hence thesis by A2,A3,SEQ_4:44;
end;
