reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th32:
  for T being connected TopSpace, A being closed open Subset of T
  holds A = {} or A = [#]T
proof
  let T be connected TopSpace, A be closed open Subset of T;
  assume that
A1: A <> {} and
A2: A <> [#]T;
A3: A` <> {} by A2,PRE_TOPC:4;
  A misses A` by SUBSET_1:24;
  then
A4: A, A` are_separated by CONNSP_1:2;
A5: [#]T = A \/ A` by PRE_TOPC:2;
  A <> {}T by A1;
  then not [#]T is connected by A5,A4,A3,CONNSP_1:15;
  hence thesis by CONNSP_1:27;
end;
