
theorem Th34:
  for a, b being Point of I[01] st a < b & a <> 0 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: a <> 0;
A3: b <= 1 by BORSUK_1:43;
  [. 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 7;
    then consider r being Real such that
A4: x = r and
A5: a <= r and
A6: r < b;
A7: r < 1 by A3,A6,XXREAL_0:2;
    0 < r by A2,A5,BORSUK_1:43;
    hence thesis by A4,A7,Th32;
  end;
  hence thesis by A1,XXREAL_1:3;
end;
