reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem
  f|A is bounded_above & i in dom D implies (upper_bound rng f)*vol(
  divset(D,i)) >= (upper_bound rng (f|divset(D,i)))*vol(divset(D,i))
proof
A1: dom f = A by FUNCT_2:def 1;
  assume f|A is bounded_above;
  then
A2: rng f is bounded_above by Th11;
  assume i in dom D;
  then dom (f|divset(D,i)) = divset(D,i) by A1,Th6,RELAT_1:62;
  then
A3: rng(f|divset(D,i)) is non empty Subset of REAL by RELAT_1:42;
A4: 0 <= vol(divset(D,i)) by SEQ_4:11,XREAL_1:48;
  rng(f|divset(D,i)) c= rng f by RELAT_1:70;
  hence thesis by A3,A2,A4,SEQ_4:48,XREAL_1:64;
end;
