reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem
  Component_of x c= qComponent_of x
proof
  consider F9 being Subset-Family of X such that
A1: for A being Subset of X holds (A in F9 iff A is open closed & x in A ) and
A2: qComponent_of x = meet F9 by Def7;
A3: for B1 being set st B1 in F9 holds Component_of x c= B1
  proof
    set S=Component_of x;
    let B1 be set;
A4: x in S by CONNSP_1:38;
    assume
A5: B1 in F9;
    then reconsider B=B1 as Subset of X;
A6: x in B by A1,A5;
A7: B is open closed by A1,A5;
    then B` is closed;
    then Cl B` = B` by PRE_TOPC:22;
    then
A8: B misses Cl B` by XBOOLE_1:79;
A9: S /\ B c= B & S /\ B` c= B` by XBOOLE_1:17;
    Cl B = B by A7,PRE_TOPC:22;
    then Cl B misses B` by XBOOLE_1:79;
    then B,B` are_separated by A8,CONNSP_1:def 1;
    then
A10: S /\ B,S /\ B` are_separated by A9,CONNSP_1:7;
A11: [#] X = B \/ B` by PRE_TOPC:2;
    S = S /\ [#] X by XBOOLE_1:28
      .= (S /\ B) \/ (S /\ B`) by A11,XBOOLE_1:23;
    then S /\ B = {}X or S /\ B` = {}X by A10,CONNSP_1:15;
    then S misses B` by A6,A4,XBOOLE_0:def 4,def 7;
    then S c= B`` by SUBSET_1:23;
    hence thesis;
  end;
  F9 <> {} by A1,Th22;
  hence thesis by A2,A3,SETFAM_1:5;
end;
