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

theorem Th10:
  for C being non empty Subset of TopSpaceMetr M, p being Point of
M st for q being Point of M st q in C holds dist(p,q) >= r holds (dist_min C).p
  >= r
proof
  let C be non empty Subset of TopSpaceMetr M, p be Point of M such that
A1: for q being Point of M st q in C holds dist(p,q) >= r;
  set B = [#]((dist p).:C);
A2: B = (dist p).:C by WEIERSTR:def 1;
A3: TopSpaceMetr M = TopStruct (#the carrier of M,Family_open_set M#) by
PCOMPS_1:def 5;
A4: for s being Real st s in B holds r <= s
  proof
    let s be Real;
    assume s in B;
    then consider y being object such that
    y in dom dist p and
A5: y in C and
A6: s = (dist p).y by A2,FUNCT_1:def 6;
    reconsider y9 = y as Point of M by A3,A5;
    s = dist(p,y9) by A6,WEIERSTR:def 4;
    hence thesis by A1,A5;
  end;
  dom dist p = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  then lower_bound B >= r by A2,A4,SEQ_4:43;
  then lower_bound((dist p).:C) >= r by WEIERSTR:def 3;
  hence thesis by WEIERSTR:def 6;
end;
