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 Th33:
  f|A is bounded & f is integrable & g|A is bounded & g is
integrable implies f-g is integrable & integral(f-g) = integral(f)-integral(g)
proof
  assume that
A1: f|A is bounded & f is integrable and
A2: g|A is bounded and
A3: g is integrable;
A4: -g is integrable by A2,A3,Th31;
A5: (-g)|A is bounded by A2,RFUNCT_1:82;
  hence f-g is integrable by A1,A4,INTEGRA1:57;
  integral(-g) = (-1)*integral(g) by A2,A3,Th31;
  then integral(f-g)=integral(f)+-integral(g) by A1,A5,A4,INTEGRA1:57;
  hence thesis;
end;
