theorem Th34:
  for P being Subset of I[01] st
  P=(the carrier of I[01]) \{0,1} holds P is open
proof
  let P be Subset of I[01];
  assume
A1: P=(the carrier of I[01]) \{0,1};
A2: 0 in [.0,1.] by XXREAL_1:1;
A3: 1 in [.0,1.] by XXREAL_1:1;
  reconsider Q0={0} as Subset of I[01] by A2,BORSUK_1:40,ZFMISC_1:31;
  reconsider Q1={1} as Subset of I[01] by A3,BORSUK_1:40,ZFMISC_1:31;
  Q0 \/ Q1 is closed by A2,A3,BORSUK_1:40,TOPS_1:9;
  then [#](I[01])\(Q0 \/ Q1) is open;
  hence thesis by A1,ENUMSET1:1;
end;
