reserve th, th1, th2, th3 for Real;

theorem Th37:
  sin(th1+th2)*sin(th1-th2) = sin(th1)*sin(th1)-sin(th2)*sin(th2)
proof
  sin(th1+th2)*sin(th1-th2) = (sin(th1)*cos(th2)+cos(th1)*sin(th2))*sin(
  th1-th2) by SIN_COS:75
    .= (sin(th1)*cos(th2)+cos(th1)*sin(th2)) *(sin(th1)*cos(th2)---cos(th1)*
  sin(th2)) by SIN_COS:82
    .= ((sin(th1)*sin(th1)*(cos(th2)*cos(th2))) -(cos(th1)*sin(th2)*cos(th1)
  )*sin(th2))
    .= (sin(th1)*sin(th1)*(1-sin(th2)*sin(th2)) -((cos(th1)*cos(th1))*sin(
  th2))*sin(th2)) by Th5
    .= (sin(th1)*sin(th1)*(1+-sin(th2)*sin(th2)) -((1---sin(th1)*sin(th1))*
  sin(th2))*sin(th2)) by Th5
    .= (sin(th1)*sin(th1)-sin(th2)*sin(th2));
  hence thesis;
end;
