reserve n for Nat;

theorem
  for C being Simple_closed_curve, i being Nat holds Fr (UBD
  L~Cage (C,i))` = L~Cage (C,i)
proof
  let C be Simple_closed_curve, i be Nat;
  set K = (UBD L~Cage (C,i))`;
  set R = L~Cage (C,i);
A1: (BDD R) \/ (BDD R)` = [#] TOP-REAL 2 by PRE_TOPC:2;
  UBD R c= R` by JORDAN2C:26;
  then
A2: UBD R misses R by SUBSET_1:23;
  UBD R misses BDD R by JORDAN2C:24;
  then
A3: UBD R misses (BDD R) \/ R by A2,XBOOLE_1:70;
  [#] TOP-REAL 2 = R` \/ R by PRE_TOPC:2
    .= (BDD R) \/ (UBD R) \/ R by JORDAN2C:27;
  then
A4: K = ((UBD R) \/ ((BDD R) \/ R)) \ UBD R by XBOOLE_1:4
    .= R \/ BDD R by A3,XBOOLE_1:88;
  ((BDD R) \/ (UBD R))` = R`` by JORDAN2C:27;
  then (BDD R)` /\ (UBD R)` = R by XBOOLE_1:53;
  then (BDD R) \/ R = ((BDD R) \/ (BDD R)`) /\ ((BDD R) \/ K) by XBOOLE_1:24
    .= (BDD R) \/ K by A1,XBOOLE_1:28
    .= K by A4,XBOOLE_1:7,12;
  then
A5: Cl K = (BDD L~Cage (C,i)) \/ L~Cage (C,i) by PRE_TOPC:22;
A6: K` = LeftComp Cage (C,i) by GOBRD14:36;
  BDD L~Cage (C,i) misses UBD L~Cage (C,i) by JORDAN2C:24;
  then
A7: (BDD L~Cage (C,i)) /\ (UBD L~Cage (C,i)) = {};
  Fr K = Cl K /\ Cl K` by TOPS_1:def 2
    .= ((BDD L~Cage (C,i)) \/ L~Cage (C,i)) /\ ((UBD L~Cage (C,i)) \/ L~Cage
  (C,i)) by A5,A6,GOBRD14:22
    .= ((BDD L~Cage (C,i)) /\ (UBD L~Cage (C,i))) \/ L~Cage (C,i) by
XBOOLE_1:24
    .= L~Cage (C,i) by A7;
  hence thesis;
end;
