Puzzle #219: Still More Marbles

You have some marbles, each of one colour of red, orange, yellow, green, blue, and violet.

The number of green marbles is one-quarter of the number of all of the other marbles.
The number of blue marbles is the square root of the number of orange marbles. (A square root of a number is a number that multiplied by itself equals the original number. The square root of 4 is 2 since 2 × 2 = 4.)
The number of red marbles is the cube of the number of violet marbles. (27 is the cube of 3 as 3 × 3 × 3 = 27.)
The six numbers of marbles are all different, and each is in the range of one to nine.

Given the above clues, how many of each colour do you have?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, August 16, 2017 at noon Pacific Time. I will post the answer shortly after.