
theorem Th2:
  dyadic(0) = {0,1}
proof
A1: 2|^0 = 1 by NEWTON:4;
A2: for x being Real
     holds x in dyadic(0) iff ex i being Nat st
    i <= 1 & x = i
  proof
    let x be Real;
A3: (ex i being Nat st i <= 1 & x = i) implies x in
    dyadic(0)
    proof
      given i being Nat such that
A4:   i <= 1 and
A5:   x = i;
      x = i/1 by A5;
      hence thesis by A1,A4,Def1;
    end;
    x in dyadic(0) implies ex i being Nat st i <= 1 & x = i
    proof
      assume x in dyadic(0);
      then ex i being Nat st i <= 1 & x = i/1 by A1,Def1;
      hence thesis;
    end;
    hence thesis by A3;
  end;
  for x being object holds x in dyadic(0) iff x in {0,1}
  proof
    let x be object;
A6: x in dyadic(0) implies x in {0,1}
    proof
      assume
A7:   x in dyadic(0);
      then reconsider x as Real;
      consider i being Nat such that
A8:   i <= 1 and
A9:  x = i by A2,A7;
      i <= 0+1 by A8;
      then x = 0 or x = 1 by A9,NAT_1:9;
      hence thesis by TARSKI:def 2;
    end;
    x in {0,1} implies x in dyadic(0)
    proof
      assume x in {0,1};
      then x = 0 or x = 1 by TARSKI:def 2;
      hence thesis by A2;
    end;
    hence thesis by A6;
  end;
  hence thesis by TARSKI:2;
end;
