reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;

theorem Th9:
  for C being non empty Subset of TopSpaceMetr M, p being Point of
  TopSpaceMetr M holds (dist_min C).p >= 0
proof
  let C be non empty Subset of TopSpaceMetr M, p be Point of TopSpaceMetr M;
A1: TopSpaceMetr M = TopStruct (#the carrier of M,Family_open_set M#) by
PCOMPS_1:def 5;
  then reconsider x = p as Point of M;
  set B = [#]((dist x).:C);
A2: B = (dist x).:C by WEIERSTR:def 1;
A3: for r being Real st r in B holds 0 <= r
  proof
    let r be Real;
    assume r in B;
    then consider y being object such that
    y in dom dist x and
A4: y in C and
A5: r = (dist x).y by A2,FUNCT_1:def 6;
    reconsider y9 = y as Point of M by A1,A4;
    r = dist(x,y9) by A5,WEIERSTR:def 4;
    hence thesis by METRIC_1:5;
  end;
  dom dist x = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  then lower_bound B >= 0 by A2,A3,SEQ_4:43;
  then lower_bound((dist x).:C) >= 0 by WEIERSTR:def 3;
  hence thesis by WEIERSTR:def 6;
end;
