reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

theorem Th41:
  for D being Subset of TOP-REAL n, A,C being non empty Subset of
  TOP-REAL n st C c= D holds dist_min(A,D) <= dist_min(A,C)
proof
  let D be Subset of TOP-REAL n;
  let A,C be non empty Subset of TOP-REAL n such that
A1: C c= D;
  consider A9,D9 be Subset of TopSpaceMetr Euclid n such that
A2: A = A9 and
A3: D = D9 & dist_min(A,D) = min_dist_min(A9,D9) by Def1;
  reconsider f = dist_min A9 as Function of the carrier of TopSpaceMetr Euclid
  n, REAL by TOPMETR:17;
  reconsider Y = f.:D9 as Subset of REAL;
A4: Y is bounded_below
  proof
    take 0;
    let r be ExtReal;
    assume r in Y;
    then ex c being Element of TopSpaceMetr Euclid n st c in D9 & r = f.c by
FUNCT_2:65;
    hence thesis by A2,Th9;
  end;
A5: lower_bound Y = lower_bound([#]((dist_min A9).:D9)) by WEIERSTR:def 1
    .= lower_bound((dist_min A9).:D9) by WEIERSTR:def 3
    .= min_dist_min(A9,D9) by WEIERSTR:def 7;
  consider A99,C9 be Subset of TopSpaceMetr Euclid n such that
A6: A = A99 and
A7: C = C9 and
A8: dist_min(A,C) = min_dist_min(A99,C9) by Def1;
  dom f = the carrier of TopSpaceMetr Euclid n by FUNCT_2:def 1;
  then reconsider X = f.:C9 as non empty Subset of REAL by A7;
  lower_bound X = lower_bound([#]((dist_min A9).:C9)) by WEIERSTR:def 1
    .= lower_bound((dist_min A9).:C9) by WEIERSTR:def 3
    .= min_dist_min(A9,C9) by WEIERSTR:def 7;
  hence thesis by A1,A2,A3,A6,A7,A8,A5,A4,RELAT_1:123,SEQ_4:47;
end;
