reserve th, th1, th2, th3 for Real;

theorem Th4:
  sin(th)*sin(th) = 1-cos(th)*cos(th)
proof
  sin(th)*sin(th) = sin(th)*sin(th)+(1-1)
    .= sin(th)*sin(th)+(1---(sin(th)*sin(th)+cos(th)*cos(th))) by SIN_COS:29
    .= 1-cos(th)*cos(th);
  hence thesis;
end;
