reserve a,b,e,r,x,y for Real,
  i,j,k,n,m for Element of NAT,
  x1 for set,
  p,q for FinSequence of REAL,
  A for non empty closed_interval Subset of REAL,
  D,D1,D2 for Division of A,
  f,g for Function of A,REAL,
  T for DivSequence of A;

theorem Th19:
  f|A is bounded & i in dom D implies upper_bound rng(f|divset(D,i
  )) >= lower_bound rng f
proof
  assume
A1: f|A is bounded;
  assume i in dom D;
  then divset(D,i) c= A by INTEGRA1:8;
  hence thesis by A1,Lm4;
end;
