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

theorem Th13:
  for A,B being non empty Subset of TopSpaceMetr M st for p,q
being Point of M st p in A & q in B holds dist(p,q) >= r holds min_dist_min(A,B
  ) >= r
proof
  let A,B be non empty Subset of TopSpaceMetr M such that
A1: for p,q being Point of M st p in A & q in B holds dist(p,q) >= r;
  set X = [#]((dist_min A).:B);
A2: X = (dist_min A).:B by WEIERSTR:def 1;
A3: for s being Real st s in X holds r <= s
  proof
    let s be Real;
    assume s in X;
    then consider y being object such that
    y in dom dist_min A and
A4: y in B and
A5: s = (dist_min A).y by A2,FUNCT_1:def 6;
    reconsider y as Point of TopSpaceMetr M by A4;
    reconsider p = y as Point of M by TOPMETR:12;
    for q being Point of M st q in A holds dist(p,q) >= r by A1,A4;
    hence thesis by A5,Th10;
  end;
  dom dist_min A = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  then lower_bound X >= r by A2,A3,SEQ_4:43;
  then lower_bound((dist_min A).:B) >= r by WEIERSTR:def 3;
  hence thesis by WEIERSTR:def 7;
end;
