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

theorem Th11:
  for A,B being non empty Subset of TopSpaceMetr M holds min_dist_min(A,B) >= 0
proof
  let A,B be non empty Subset of TopSpaceMetr M;
  set X = [#]((dist_min A).:B);
A1: X = (dist_min A).:B by WEIERSTR:def 1;
A2: for r being Real st r in X holds 0 <= r
  proof
    let r be Real;
    assume r in X;
    then consider y being object such that
    y in dom dist_min A and
A3: y in B and
A4: r = (dist_min A).y by A1,FUNCT_1:def 6;
    reconsider y as Point of TopSpaceMetr M by A3;
    (dist_min A).y >= 0 by Th9;
    hence thesis by A4;
  end;
  dom dist_min A = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  then lower_bound X >= 0 by A1,A2,SEQ_4:43;
  then lower_bound((dist_min A).:B) >= 0 by WEIERSTR:def 3;
  hence thesis by WEIERSTR:def 7;
end;
