reserve X, Y, Z, W for non empty MetrSpace;
reserve X,Y for non empty MetrSpace;
reserve Z for non empty MetrSpace;

theorem Th16: :::
  for a,b,c,d,e,f being Real holds (2*(a*d)*(c*b) + 2*(a*f)
*(e*c) + 2*(b*f)*(e*d)) <= (((a*d)^2 + (c*b)^2 + (a*f)^2 + (e*c)^2 + (b*f)^2) +
  (e*d)^2)
proof
  let a,b,c,d,e,f be Real;
  0 <= ((a*f) - (e*c))^2 & 0 <= ((b*f) - (e*d))^2 by XREAL_1:63;
  then
  0 <= ((a*d) - (c*b))^2 & 0 + 0 <= ((a*f) - (e*c))^2 + ((b*f) - (e*d))^2
  by XREAL_1:7,63;
  then
  0 + 0 <= ((a*d) - (c*b))^2 + (((a*f) - (e*c))^2 + ((b*f) - (e*d))^2) by
XREAL_1:7;
  then
  0 <= (((a*d)^2 + (c*b)^2) + ((a*f) - (e*c))^2 + ((b*f) - (e*d))^2) - 2*(
  a*d)*(c*b);
  then
  0 + 2*(a*d)*(c*b) <= ((a*d)^2 + (c*b)^2) + ((a*f) - (e*c))^2 + ((b*f) -
  (e*d))^2 by XREAL_1:19;
  then
  2*(a*d)*(c*b) <= ((a*d)^2 + (c*b)^2 + (a*f)^2 + (e*c)^2 + ((b*f) - (e*d)
  )^2) - 2*(a*f)*(e*c);
  then
  (2*(a*d)*(c*b) + 2*(a*f)*(e*c)) <= (((a*d)^2 + (c*b)^2 + (a*f)^2 + (e*c)
  ^2 + (b*f)^2) + (e*d)^2) - 2*(b*f)*(e*d) by XREAL_1:19;
  hence thesis by XREAL_1:19;
end;
