reserve k,m,n for Nat,
  i1,i2,i3 for Integer,
  e for set;
reserve i,k,m,n,p,x,y for Nat;
reserve a for Tuple of n,(k-SD);

theorem Th20:
  m is_represented_by 1,k implies DigA(DecSD(m,1,k),1) = m
proof
  assume m is_represented_by 1,k;
  then
A1: m < Radix(k) |^ 1;
  1 in Seg 1 by FINSEQ_1:1;
  hence DigA(DecSD(m,1,k),1) = DigitDC(m,1,k) by Def9
    .= (m mod (Radix(k) |^ 1)) div (Radix(k) |^ 0) by XREAL_1:232
    .= (m mod (Radix(k) |^ 1)) div 1 by NEWTON:4
    .= m mod (Radix(k) |^ 1) by NAT_2:4
    .= m by A1,NAT_D:24;
end;
