reserve i, j, k, n for Nat,
  P for Subset of TOP-REAL 2,
  C for connected compact non vertical non horizontal Subset of TOP-REAL 2;

theorem
  meet BDD-Family C = C \/ BDD C
proof
  thus meet BDD-Family C c= C \/ BDD C
  proof
    let x be object;
    assume
A1: x in meet BDD-Family C;
    COMPLEMENT UBD-Family C = BDD-Family C by Th20;
    then
    ([#] the carrier of TOP-REAL 2) \ union UBD-Family C = meet BDD-Family
    C by SETFAM_1:33;
    then not x in union UBD-Family C by A1,XBOOLE_0:def 5;
    then
A2: not x in UBD C by Th14;
    per cases;
    suppose
A3:   not x in C;
A4:   BDD C \/ UBD C = C` by JORDAN2C:27;
      x in C` by A1,A3,SUBSET_1:29;
      then x in BDD C by A2,A4,XBOOLE_0:def 3;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      x in C;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  (BDD C) misses (UBD C) by JORDAN2C:24;
  then
A5: (BDD C) /\ (UBD C) = {};
A6: BDD C c= meet BDD-Family C
  proof
    let x be object;
    assume
A7: x in BDD C;
    then not x in UBD C by A5,XBOOLE_0:def 4;
    then
A8: not x in union UBD-Family C by Th14;
    for Y being set st Y in BDD-Family C holds x in Y
    proof
      let Y be set;
      assume Y in BDD-Family C;
      then consider n such that
A9:   Y = Cl BDD L~Cage(C,n) and
      not contradiction;
      LeftComp Cage(C,n) is_outside_component_of L~Cage(C,n) by GOBRD14:34;
      then
A10:  LeftComp Cage(C,n) in {B where B is Subset of TOP-REAL 2: B
      is_outside_component_of L~Cage(C,n)};
      BDD C misses L~Cage(C,n) by Th19;
      then BDD C /\ L~Cage(C,n) = {};
      then
A11:  not x in L~Cage(C,n) by A7,XBOOLE_0:def 4;
      RightComp Cage(C,n) is_inside_component_of L~Cage(C,n) by GOBRD14:35;
      then
A12:  RightComp Cage(C,n) in {B where B is Subset of TOP-REAL 2: B
      is_inside_component_of L~Cage(C,n)};
      UBD L~Cage(C,n) in UBD-Family C;
      then UBD L~Cage(C,n) = union{B where B is Subset of TOP-REAL 2: B
      is_outside_component_of L~Cage(C,n)} & not x in UBD L~Cage(C,n) by A8,
JORDAN2C:def 5,TARSKI:def 4;
      then not x in LeftComp Cage(C,n) by A10,TARSKI:def 4;
      then BDD L~Cage(C,n) = union{B where B is Subset of TOP-REAL 2: B
      is_inside_component_of L~Cage(C,n)} & x in RightComp Cage(C,n) by A7,A11,
GOBRD14:18,JORDAN2C:def 4;
      then
A13:  x in BDD L~Cage(C,n) by A12,TARSKI:def 4;
      BDD L~Cage(C,n) c= Cl BDD L~Cage(C,n) by PRE_TOPC:18;
      hence thesis by A9,A13;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
  C c= meet BDD-Family C by Th15;
  hence thesis by A6,XBOOLE_1:8;
end;
