reserve a,b,r,x,y for Real,
  i,j,k,n for Nat,
  x1 for set;
reserve A, B for non empty closed_interval Subset of REAL;
reserve f, g for Function of A,REAL;
reserve D, D1, D2 for Division of A;

theorem
  f|A is bounded & f is integrable & g|A is bounded & g is integrable &
  (for x st x in A holds f.x >= g.x) implies integral(f) >= integral(g)
proof
  assume that
A1: f|A is bounded & f is integrable & g|A is bounded & g is integrable and
A2: for x st x in A holds f.x >= g.x;
A3: dom (f-g)= A by FUNCT_2:def 1;
A4: for x st x in A holds (f-g).x >= 0
  proof
    let x;
    assume
A5: x in A;
    then (f-g).x = f.x - g.x by A3,VALUED_1:13;
    hence thesis by A2,A5,XREAL_1:48;
  end;
  integral(f-g)=integral(f)-integral(g) & (f-g)|(A /\ A) is bounded by A1,Th33,
RFUNCT_1:84;
  hence thesis by A4,Th32,XREAL_1:49;
end;
