reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th68:
  f is_differentiable_on Z & A c= Z & f`|Z is_integrable_on A & (f
`|Z)|A is bounded implies integral((r(#)(f`|Z)),A) = r*f.(upper_bound A) - r*f.
  (lower_bound A)
proof
  assume that
A1: f is_differentiable_on Z & A c= Z and
A2: f`|Z is_integrable_on A & (f`|Z)|A is bounded;
A3: (f`|Z)||A is integrable & (f`|Z)||A|A is bounded by A2,INTEGRA5:9;
  A c= dom(f`|Z) by A1,FDIFF_1:def 7;
  then
A4: (f`|Z)||A is Function of A,REAL by FUNCT_2:68,INTEGRA5:6;
  integral((r(#)(f`|Z)),A) = integral (r(#)((f`|Z)||A)) by RFUNCT_1:49
    .= r * integral ((f`|Z),A) by A4,A3,INTEGRA2:31
    .= r*(f.(upper_bound A) - f.(lower_bound A)) by A1,A2,INTEGRA5:13;
  hence thesis;
end;
