
theorem Th12:
  for A,B be non empty closed_interval Subset of REAL st
    B c= A holds
  lower_bound A <= lower_bound B & 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 XXREAL_1:50,A2;
end;
