reserve a, r, s for Real;

theorem
  for S, T being TopSpace st the TopStruct of S = the TopStruct of T & S
  is connected holds T is connected
proof
  let S, T be TopSpace such that
A1: the TopStruct of S = the TopStruct of T and
A2: S is connected;
  let A, B be Subset of T such that
A3: [#]T = A \/ B and
A4: A,B are_separated;
  reconsider A1 = A, B1 = B as Subset of S by A1;
  [#]S = the carrier of S & A1,B1 are_separated by A1,A4,Th5;
  then A1 = {}S or B1 = {}S by A1,A2,A3;
  hence thesis;
end;
