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 Th15:
  not(C c= BDD D & D c= BDD C)
proof
  assume that
A1: C c= BDD D and
A2: D c= BDD C;
  UBD C meets UBD D by Th12;
  then consider e being object such that
A3: e in UBD C and
A4: e in UBD D by XBOOLE_0:3;
  reconsider p = e as Point of TOP-REAL 2 by A3;
  UBD D misses BDD D by JORDAN2C:24;
  then
A5: not p in BDD D by A4,XBOOLE_0:3;
  UBD C misses BDD C by JORDAN2C:24;
  then
A6: not p in BDD C by A3,XBOOLE_0:3;
  then dist(p,C) <= dist(p, BDD C) by Th14;
  then dist(p, BDD D) < dist(p, BDD C) by A1,A5,JORDAN1K:51,XXREAL_0:2;
  then dist(p, BDD D) < dist(p,D) by A2,A6,JORDAN1K:51,XXREAL_0:2;
  hence contradiction by A5,Th14;
end;
