
theorem
  for a,b,c,d be positive Real st (a + b <= c + d & a*b > c*d) holds
  max (a,b) < max(c,d) & min (a,b) > min(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) & c * d = max (c,d) * min (c,d)
      by NEWTON04:18;
    assume
    A2: (a + b <= c + d & a*b > c*d); then
    max (a,b) <= max (c,d) by SMI; then
    per cases by XXREAL_0:1;
    suppose
      B1: max (a,b) = max (c,d); then
      min (a,b) <= min (c,d) by A1,A2,XREAL_1:6;
      hence thesis by A2,B1,A1,XREAL_1:64;
    end;
    suppose
      max (a,b) < max (c,d);
      hence thesis by A1,A2,XREAL_1:66;
    end;
  end;
