
theorem Th33:
  for a, b being Point of I[01] st a < b & b <> 1 holds ]. a, b .]
  is non empty Subset of I(01)
proof
  let a, b be Point of I[01];
  assume that
A1: a < b and
A2: b <> 1;
  b <= 1 by BORSUK_1:43;
  then
A3: b < 1 by A2,XXREAL_0:1;
  ]. a, b .] c= the carrier of I(01)
  proof
    let x be object;
    assume x in ]. a, b .];
    then x in { r where r is Real: a < r & r <= b } by RCOMP_1:def 8;
    then consider r being Real such that
A4: x = r and
A5: a < r and
A6: r <= b;
A7: 0 < r by A5,BORSUK_1:43;
    r < 1 by A3,A6,XXREAL_0:2;
    hence thesis by A4,A7,Th32;
  end;
  hence thesis by A1,XXREAL_1:2;
end;
