Transitivity of discriminant of field extentions

Question

I was reading about relative discriminant of field extentions and i found a statement that for tower of extentions of fields $F \subset F' \subset F'' $ and their integel closures $A \subset A' \subset A''$ relative discriminant can be counted the following way $$ d(A'' / A) = Norm_{F':F}(d(A'':A))d(A' / A)^{[F'':F']} $$ I have calculated that for field extentnions $K \subset K' \subset K''$ the discriminant can be calculated: $$ d(F'':F) = Norm_{F':F}(d(F'':F'))d(F':F)^{[F'':F']} $$ Can i proof my statement using this formula and if i can, how i should do it? Or is there any other proof of this formula? May be direct one, or using something else.