reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th37:
  for a,b being Real,
   A being non empty closed_interval Subset of REAL st A=
  [.a,b.] holds upper_bound A=b & lower_bound A=a
proof
  let a,b be Real, A be non empty closed_interval Subset of REAL;
A1: A=[. lower_bound A, upper_bound A .] by INTEGRA1:4;
  assume A=[.a,b.];
  hence thesis by A1,INTEGRA1:5;
end;
