i already knew this but when i saw the OP i wondered why that was and its actually pretty obvious and can b proven simply.
if u write all ur numbers in base x (OP says x=10 which is ofc the most commonly used base) and then use timetables of x1 (OP says x1=9) and then write out all the timetables u get
1(x1)=(x1) obviously cuz x is the base the digit on the left will b 0 and the digit on the right will b (x1)
2(x1)=(2x2) obviously cuz x is the base the digit on the left will b 1 and the digit on the right will b (x2)
3(x1)=(3x3) obviously cuz x is the base the digit on the left will b 2 and the digit on the right will b (x3)
4(x1)=(4x4) obviously cuz x is the base the digit on the left will b 3 and the digit on the right will b (x4)
and this can b repeated for all two digit numbers when multiplying a number which is 1 under the base.
so eg if we skip to 7 times 9. so here were saying x=10 and (x1)=9
and so 7 times 9=7(x1)=7x7 which means the left hand digit is 6 and the right hand digit is x7
so we know the left hand digit is 6 and we now need to find out the right hand digit. right hand digit is (x7) and substituting (x=10) gives us (107) which is 3. so the right hand digit is 3. putting these 2 facts together gives us the number 63.
therefore 7 times 9=63
