reserve i,j,k,n for Nat,
  C for being_simple_closed_curve Subset of TOP-REAL 2;
reserve p,q for Point of TOP-REAL 2,
  D for Simple_closed_curve;

theorem Th14:
  not p in BDD C implies dist(p,C) <= dist(p,BDD C)
proof
  per cases;
  suppose
    p in C;
    then dist(p,C) = 0 by JORDAN1K:45;
    hence thesis by JORDAN1K:44;
  end;
  suppose
A1: not p in C;
    assume that
A2: not p in BDD C and
A3: dist(p,C) > dist(p,BDD C);
A4: ex q st q in BDD C & dist(p,q) < dist(p,C) by A3,JORDAN1K:48;
    p in C` by A1,SUBSET_1:29;
    then p in (BDD C) \/ (UBD C) by JORDAN2C:27;
    then p in UBD C by A2,XBOOLE_0:def 3;
    hence contradiction by A4,Th13;
  end;
end;
