reserve k for Nat;

theorem Th14:
  for x,n,k be Nat holds DecSD2(x,n,k) = DecSD(x,n,k)
proof
  let x,n,k be Nat;
A1: len DecSD2(x,n,k) = n by CARD_1:def 7;
  then
A2: dom DecSD2(x,n,k) = Seg n by FINSEQ_1:def 3;
A3: now
    let j be Nat;
    assume
A4: j in dom DecSD2(x,n,k);
    then DecSD2(x,n,k).j = DigitDC2(x,j,k) by A2,Def5
      .= DigitDC(x,j,k) by RADIX_1:def 8
      .= DigA(DecSD(x,n,k),j) by A2,A4,RADIX_1:def 9
      .= DecSD(x,n,k).j by A2,A4,RADIX_1:def 3;
    hence DecSD2(x,n,k).j = DecSD(x,n,k).j;
  end;
  len DecSD(x,n,k) = n by CARD_1:def 7;
  hence thesis by A1,A3,FINSEQ_2:9;
end;
