First consider case where the input is only one digit: superdigit(x where x<10)

its obvious that superdigit(x) =x mod 9 if x is smaller than 9. Only x==9 will be an edge case. Now we consider a decimal number such as 100*a+10*b+c and we can prove that 100*a+10*b+c mod 9 =(a+b+c) mod 9 +(99*a+9*b)mod 9 100*a+10*b+c mod 9=0+(a+b+c) mod 9 which means mod9 conserves through recursive superdigit operation.

Then if we have any given input a, we can draw a chain like a-b-c-..-x where x is a single digit number.

From the previous paragraph, we know that amod9==bmod9==cmod9 .... ==xmod9

From the property defined in superdigit operation, we know that supdigit(a)==superdigit(b)==superdigit(c)==...superdigit(x)

Since in the first pragraph, we showed that superdigit(x)==xmod9 despite the edge case where x==9. This tells us that tails of two chains are equal, so the heads of two chains must also be equal except x==9 case.

Therefore supdigit(a)==a mod 9 if a mod 9!=0 supdigit(a)==9 if a mod9 ==0

which