
theorem NN1:
  for a,b be negative Real holds (a/b + b/a)/2 >= 1
  proof
    let a,b be negative 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_1:72; 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;
