reserve i,j,k,n for Nat,
  D for non empty set,
  f, g for FinSequence of D;

theorem Th1:
  for T being non empty TopSpace for B,C1,C2,D being Subset of T st
B is connected & C1 is_a_component_of D & C2 is_a_component_of D & B meets C1 &
  B meets C2 & B c= D holds C1 = C2
proof
  let T be non empty TopSpace;
  let B,C1,C2,D be Subset of T;
  assume that
A1: B is connected and
A2: C1 is_a_component_of D & C2 is_a_component_of D and
A3: B meets C1 and
A4: B meets C2 & B c= D;
A5: B <> {} by A3,XBOOLE_1:65;
  B c= C1 & B c= C2 by A1,A2,A3,A4,GOBOARD9:4;
  hence thesis by A2,A5,GOBOARD9:1,XBOOLE_1:68;
end;
