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 = [.2*n*PI,(2*n+1)*PI.] implies integral(cos,A) = 0
proof
  assume A=[.2*n*PI,(2*n+1)*PI.];
  then upper_bound A=(2*n+1)*PI & lower_bound A=2*n*PI by Th37;
  then integral(cos,A) = sin(0+(2*n+1)*PI) - sin(0+2*n*PI) by Th39
    .= -sin(0) - sin(0+2*n*PI) by Th2
    .= -sin(0) - sin(0) by Th1
    .= -sin(0+2 * PI) - sin(0) by SIN_COS:79
    .= 0 - 0 by SIN_COS:77,79;
  hence thesis;
end;
