
theorem
  for A,B be non empty closed_interval Subset of REAL
  st B c< A holds
    lower_bound A < lower_bound B or
    upper_bound B < upper_bound A
proof
 let A,B be non empty closed_interval Subset of REAL;
 assume B c< A; then
 [. lower_bound B, upper_bound B .] c< A by INTEGRA1:4; then
 A2:[. lower_bound B, upper_bound B .]  c< [. lower_bound A, upper_bound A .]
    by INTEGRA1:4;
 lower_bound B <= upper_bound B by SEQ_4:11;
 hence thesis by A2,XXREAL_1:82;
end;
