reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem
  A = [.-2*PI,2*PI.] implies sin is_orthogonal_with cos,A
proof
  assume A = [.-2*PI,2*PI.];
  then
A1: upper_bound A = 2*PI & lower_bound A = -2*PI by INTEGRA8:37;
  |||(sin,cos,A)||| = 1/2*(cos.(lower_bound A)*cos.(lower_bound A) -cos.(
  upper_bound A)*cos.(upper_bound A)) by INTEGRA8:90
    .= 1/2*(cos.(2*PI)*cos.(-2*PI) -cos.(2*PI)*cos.(2*PI)) by A1,SIN_COS:30
    .= 1/2*(cos.(2*PI)*cos.(2*PI) -cos.(2*PI)*cos.(2*PI)) by SIN_COS:30;
  hence thesis;
end;
