
theorem SMI:
  for a,b,c,d be positive Real st a + b <= c + d & max (a,b) > max (c,d)
  holds a*b < c*d
  proof
    let a,b,c,d be positive Real;
    A1: a + b = max (a,b) + min (a,b) & c + d = max (c,d) + min (c,d) &
    a * b = max (a,b)*min(a,b) & max (c,d)*min(c,d) = c * d by NEWTON04:18;
    assume
    A2: a + b <= c + d & max (a,b) > max (c,d); then
    A4: (max (a,b) + min (a,b))*(max (a,b) + min (a,b)) <=
    (max (c,d) + min (c,d))*(max (c,d) + min (c,d)) by A1,XREAL_1:66;
    min (a,b) < min (c,d) by A1,A2,XREAL_1:8; then
    max (a,b)*max(a,b) > max (c,d)*max (c,d) &
    min (a,b)*min(a,b) < min(c,d)*min(c,d) by A2,XREAL_1:96; then
    max (a,b)*max(a,b) - min (a,b)*min (a,b) > max (c,d)*max(c,d) -
      min (c,d)*min (c,d) by XREAL_1:14; then
    (max (a,b) - min (a,b))*(max(a,b) + min(a,b)) > (max (c,d)- min (c,d))*
      (max (c,d) + min (c,d)); then
    max (a,b) - min (a,b) > max (c,d) - min (c,d) by A1,A2,XREAL_1:66; then
    (max (a,b) - min (a,b))*(max (a,b) - min (a,b)) > (max (c,d) - min(c,d))*
      (max (c,d) - min (c,d)) by XREAL_1:96; then
    (max (a,b) + min (a,b))*(max (a,b) + min (a,b)) - (max (a,b) - min (a,b))*
      (max (a,b) - min (a,b)) <
    (max (c,d) + min (c,d))*(max (c,d) + min (c,d)) - (max (c,d) - min (c,d))*
      (max (c,d) - min (c,d)) by A4,XREAL_1:15; then
    4*(max (a,b)*min (a,b)) < 4 *(max (c,d)*min (c,d));
    hence thesis by A1,XREAL_1:64;
  end;
