
theorem
  dyadic(1) = {0,1/2,1}
proof
A1: 2|^1 = 2;
  for x being object holds x in dyadic(1) iff x in {0,1/2,1}
  proof
    let x be object;
A2: x in {0,1/2,1} implies x in dyadic(1)
    proof
      assume
A3:   x in {0,1/2,1};
      per cases by A3,ENUMSET1:def 1;
      suppose
        x = 0;
        then x = 0/2;
        hence thesis by A1,Def1;
      end;
      suppose
        x = 1/2;
        hence thesis by A1,Def1;
      end;
      suppose
A4:     x = 1;
        then reconsider x as Real;
        x = 2/2 by A4;
        hence thesis by A1,Def1;
      end;
    end;
    x in dyadic(1) implies x in {0,1/2,1}
    proof
      assume
A5:   x in dyadic(1);
      then reconsider x as Real;
      consider i being Nat such that
A6:   i <= 2 and
A7:   x = i/2 by A1,A5,Def1;
      i = 0 or ... or i = 2 by A6;
      hence thesis by A7,ENUMSET1:def 1;
    end;
    hence thesis by A2;
  end;
  hence thesis by TARSKI:2;
end;
