reserve k for Nat;

theorem Th7:
  for x be Tuple of 1,k-SD holds SDDec(x) = DigA(x,1)
proof
  1 - 1 = 0;
  then
A1: 1 -' 1 = 0 by XREAL_0:def 2;
  let x be Tuple of 1,k-SD;
A2: 1 in Seg 1 by FINSEQ_1:2,TARSKI:def 1;
  then
A3: (DigitSD(x))/.1 = SubDigit(x,1,k) by RADIX_1:def 6;
A4: len (DigitSD(x)) = 1 by CARD_1:def 7;
  then 1 in dom DigitSD(x) by A2,FINSEQ_1:def 3;
  then
A5: DigitSD(x).1 = SubDigit(x,1,k) by A3,PARTFUN1:def 6;
  thus SDDec(x) = Sum DigitSD(x) by RADIX_1:def 7
    .= Sum <*SubDigit(x,1,k)*> by A4,A5,FINSEQ_1:40
    .= SubDigit(x,1,k) by RVSUM_1:73
    .= (Radix(k) |^ 0)*DigB(x,1) by A1,RADIX_1:def 5
    .= 1*DigB(x,1) by NEWTON:4
    .= DigA(x,1) by RADIX_1:def 4;
end;
