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 Th46:
  f|A is bounded implies lower_sum(f,D1) <= upper_sum(f,D2)
proof
  consider D such that
A1: D1 <= D and
A2: D2 <= D by Th45;
  assume
A3: f|A is bounded;
  then
A4: lower_sum(f,D) <= upper_sum(f,D) by Th26;
  upper_sum(f,D) <= upper_sum(f,D2) by A3,A2,Th43;
  then
A5: lower_sum(f,D) <= upper_sum(f,D2) by A4,XXREAL_0:2;
  lower_sum(f,D1) <= lower_sum(f,D) by A3,A1,Th44;
  hence thesis by A5,XXREAL_0:2;
end;
