
theorem Th22:
  for X being Subset of I[01], X9 being Subset of REAL st X9 = X
  holds X9 is bounded_above bounded_below
proof
  let X be Subset of I[01], X9 be Subset of REAL;
  assume
A1: X9 = X;
  then for r being ExtReal st r in X9 holds r <= 1 by BORSUK_1:43;
  then 1 is UpperBound of X9 by XXREAL_2:def 1;
 hence X9 is bounded_above by XXREAL_2:def 10;
  for r being ExtReal st r in X9 holds 0 <= r by A1,BORSUK_1:43;
  then 0 is LowerBound of X9 by XXREAL_2:def 2;
 hence thesis by XXREAL_2:def 9;
end;
