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
  A = [.x+2*n*PI,x+(2*n+1)*PI.] implies integral(cos,A) = -2*sin(x)
proof
  assume A=[.x+2*n*PI,x+(2*n+1)*PI.];
  then upper_bound A=x+(2*n+1)*PI & lower_bound A=x+2*n*PI by Th37;
  then integral(cos,A) = sin(x+(2*n+1)*PI) - sin(x+2*n*PI) by Th39
    .= -sin(x) - sin(x+2*n*PI) by Th2
    .= -sin(x) - sin(x) by Th1
    .= -2*sin(x);
  hence thesis;
end;
