
theorem NP1:
  for a be negative Real, b be positive Real holds (a/b + b/a)/2 <= -1
  proof
    let a be negative Real, b be positive Real;
    A1: a*a/(a*b) = a/b & b*b/(a*b) = b/a by XCMPLX_1:91;
    (a + b)*(a + b) is non negative; then
    (a*a + 2*a*b + b*b) - 2*a*b >= 0 - 2*a*b by XREAL_1:6; then
    (a*a + b*b)/(2*a*b) <= (-2*a*b)/(2*a*b) by XREAL_1:73; then
    (a*a + b*b)/(2*a*b) <= -(2*a*b)/(2*a*b); then
    (a*a + b*b)/(2*(a*b)) <= -1 by XCMPLX_1:60; then
    (a*a + b*b)/(a*b)/2 <= -1 by XCMPLX_1:78;
    hence thesis by A1;
  end;
