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 Th78:
  integral(sin-cos,A) = (-cos).(upper_bound A) - (-cos).(
  lower_bound A) - (sin.(upper_bound A) - sin.(lower_bound A))
proof
A1: [#]REAL is open Subset of REAL;
A2: sin is_integrable_on A & sin|A is bounded by Lm19;
  cos is_integrable_on A & cos|A is bounded by Lm11;
  hence thesis by A2,A1,Th26,Th27,Th29,Th67,SIN_COS:68;
end;
