reserve k for Nat;

theorem Th10:
  for n be Nat for x be Tuple of (n+1),k-SD, xn be Tuple of n,k-SD
st (for i be Nat st i in Seg n holds x.i = xn.i) holds SDDec(xn) + (Radix(k) |^
  n)*DigA(x,n+1) = SDDec(x)
proof
  let n be Nat;
  let x be Tuple of (n+1),k-SD, xn be Tuple of n,k-SD;
  assume
A1: for i be Nat st i in Seg n holds x.i = xn.i;
  SDDec(x) = Sum DigitSD(x) by RADIX_1:def 7
    .= Sum(DigitSD(xn)^<*SubDigit(x,n+1,k)*>) by A1,Th9
    .= Sum DigitSD(xn) + SubDigit(x,n+1,k) by RVSUM_1:74
    .= Sum DigitSD(xn) + (Radix(k) |^ (n+1-'1))*DigB(x,n+1) by RADIX_1:def 5
    .= Sum DigitSD(xn) + (Radix(k) |^ n)*DigB(x,n+1) by NAT_D:34
    .= Sum DigitSD(xn) + (Radix(k) |^ n)*DigA(x,n+1) by RADIX_1:def 4;
  hence thesis by RADIX_1:def 7;
end;
