
theorem
  for T being TopSpace, D being mutually-disjoint open Subset-Family of
T, A being Subset of T, X being set st A is connected & X in D & X meets A & D
  is Cover of A holds A c= X
proof
  let T be TopSpace;
  let D be mutually-disjoint open Subset-Family of T;
  let A be Subset of T;
  let X be set such that
A1: T|A is connected and
A2: X in D and
A3: X meets A;
  consider x being object such that
A4: x in X & x in A by A3,XBOOLE_0:3;
  assume D is Cover of A;
  then
A5: A c= union D by SETFAM_1:def 11;
  let a be object;
  assume
A6: a in A;
  then consider Z being set such that
A7: a in Z and
A8: Z in D by A5,TARSKI:def 4;
  set D2 = {K /\ A where K is Subset of T: K in D & not a in K};
  D2 c= bool A
  proof
    let d be object;
       reconsider dd=d as set by TARSKI:1;
    assume d in D2;
    then ex K1 being Subset of T st d = K1 /\ A & K1 in D & not a in K1;
    then dd c= A by XBOOLE_1:17;
    hence thesis;
  end;
  then reconsider D2 as Subset-Family of T|A by PRE_TOPC:8;
  assume not a in X;
  then
A9: X /\ A in D2 by A2;
  x in X /\ A by A4,XBOOLE_0:def 4;
  then
A10: x in union D2 by A9,TARSKI:def 4;
  set D1 = {K /\ A where K is Subset of T: K in D & a in K};
  D1 c= bool A
  proof
    let d be object;
       reconsider dd=d as set by TARSKI:1;
    assume d in D1;
    then ex K1 being Subset of T st d = K1 /\ A & K1 in D & a in K1;
    then dd c= A by XBOOLE_1:17;
    hence thesis;
  end;
  then reconsider D1 as Subset-Family of T|A by PRE_TOPC:8;
A11: A c= union D1 \/ union D2
  proof
    let b be object;
    assume
A12: b in A;
    then consider Z being set such that
A13: b in Z and
A14: Z in D by A5,TARSKI:def 4;
    reconsider Z as Subset of T by A14;
A15: b in Z /\ A by A12,A13,XBOOLE_0:def 4;
    per cases;
    suppose
      a in Z;
      then Z /\ A in D1 by A14;
      then b in union D1 by A15,TARSKI:def 4;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      not a in Z;
      then Z /\ A in D2 by A14;
      then b in union D2 by A15,TARSKI:def 4;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
A16: Z /\ A in D1 by A7,A8;
A17: [#](T|A) = A by PRE_TOPC:def 5;
  D1 is open
  proof
    let P be Subset of T|A;
    assume P in D1;
    then consider K1 being Subset of T such that
A18: P = K1 /\ A and
A19: K1 in D and
    a in K1;
    K1 is open by A19,TOPS_2:def 1;
    hence thesis by A17,A18,TOPS_2:24;
  end;
  then
A20: union D1 is open by TOPS_2:19;
  D2 is open
  proof
    let P be Subset of T|A;
    assume P in D2;
    then consider K1 being Subset of T such that
A21: P = K1 /\ A and
A22: K1 in D and
    not a in K1;
    K1 is open by A22,TOPS_2:def 1;
    hence thesis by A17,A21,TOPS_2:24;
  end;
  then
A23: union D2 is open by TOPS_2:19;
  the carrier of T|A = A by PRE_TOPC:8;
  then
A24: A = union D1 \/ union D2 by A11;
  a in Z /\ A by A6,A7,XBOOLE_0:def 4;
  then union D1 <> {}(T|A) by A16,TARSKI:def 4;
  then union D1 meets union D2 by A1,A17,A20,A23,A24,A10,CONNSP_1:11;
  then consider y being object such that
A25: y in union D1 and
A26: y in union D2 by XBOOLE_0:3;
  consider Y2 being set such that
A27: y in Y2 and
A28: Y2 in D2 by A26,TARSKI:def 4;
  consider K2 being Subset of T such that
A29: Y2 = K2 /\ A and
A30: K2 in D & not a in K2 by A28;
A31: y in K2 by A27,A29,XBOOLE_0:def 4;
  consider Y1 being set such that
A32: y in Y1 and
A33: Y1 in D1 by A25,TARSKI:def 4;
  consider K1 being Subset of T such that
A34: Y1 = K1 /\ A and
A35: K1 in D & a in K1 by A33;
  y in K1 by A32,A34,XBOOLE_0:def 4;
  then K1 meets K2 by A31,XBOOLE_0:3;
  hence thesis by A35,A30,TAXONOM2:def 5;
end;
