
theorem
  for n being Nat holds
  dom dyad(n) = Seg((2|^n)+1) & rng dyad(n) = dyadic(n)
proof
  let n be Nat;
A1: dom dyad(n) = Seg((2|^n)+1) by Def4;
  for x being object holds x in rng dyad(n) iff x in dyadic(n)
  proof
    let x be object;
A2: x in rng dyad(n) implies x in dyadic(n)
    proof
      assume x in rng dyad(n);
      then consider y being object such that
A3:   y in dom (dyad(n)) and
A4:   x = (dyad(n)).y by FUNCT_1:def 3;
A5:   y in Seg((2|^n)+1) by A3,Def4;
      reconsider y as Nat by A3;
A6:   1 <= y by A5,FINSEQ_1:1;
      y <= (2|^n) + 1 by A5,FINSEQ_1:1;
      then
A7:   y - 1 <= (2|^n) + 1 - 1 by XREAL_1:13;
      consider i being Nat such that
A8:   1 + i = y by A6,NAT_1:10;
      i in NAT & x = (y-1)/(2|^n) by A3,A4,Def4,ORDINAL1:def 12;
      hence thesis by A7,A8,Def1;
    end;
    x in dyadic(n) implies x in rng (dyad(n))
    proof
      assume
A9:  x in dyadic(n);
      then reconsider x as Real;
      consider i being Nat such that
A10:  i <= 2|^n and
A11:  x = i/(2|^n) by A9,Def1;
      consider y being Nat such that
A12:  y = i + 1;
      0 + 1 <= i + 1 & i + 1 <= (2|^n) + 1 by A10,XREAL_1:6;
      then
A13:  y in Seg((2|^n)+1) by A12,FINSEQ_1:1;
      then
A14:  y in dom (dyad(n)) by Def4;
      x = (y-1)/(2|^n) by A11,A12;
      then x = (dyad(n)).y by A1,A13,Def4;
      hence thesis by A14,FUNCT_1:def 3;
    end;
    hence thesis by A2;
  end;
  hence thesis by Def4,TARSKI:2;
end;
