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

theorem Th72:
  for X being compact Subset of R^1, X9 being Subset of REAL st
  X9 = X holds X9 is bounded_above bounded_below
proof
  let X be compact Subset of R^1, X9 be Subset of REAL;
  assume X9 = X;
  then X9 is compact by JORDAN5A:25;
  then X9 is real-bounded by RCOMP_1:10;
  hence thesis by XXREAL_2:def 11;
end;
