
theorem Th13:
  for n being Nat holds for x being Element of dyadic(n
+1) holds (not x in dyadic(n) implies (axis(x)-1)/(2|^(n+1)) in dyadic(n) &
  (axis(x)+1)/(2|^(n+1)) in dyadic(n) )
proof
  let n be Nat;
  let x be Element of dyadic(n+1);
  assume
A1: not x in dyadic(n);
  thus (axis(x)-1)/(2|^(n+1)) in dyadic(n)
  proof
    consider a being Real such that
A2: a = (axis(x)-1)/(2|^(n+1));
    ex i being Nat st i <= 2|^n & a = i/(2|^n)
    proof
      consider s being Nat such that
A3:   s <= (2|^(n+1)) and
A4:   x = s/(2|^(n+1)) by Def1;
      consider k being Element of NAT such that
A5:   s = 2 * k or s = 2 * k + 1 by SCHEME1:1;
      now
        per cases by A5;
        case
A6:       s = k * 2;
          then x = (k * 2)/((2|^n)* 2) by A4,NEWTON:6;
          then
A7:       x = k/(2|^n) by XCMPLX_1:91;
          k * 2 <= (2|^n)* 2 by A3,A6,NEWTON:6;
          then k <= ((2|^n)* 2)/2 by XREAL_1:77;
          hence thesis by A1,A7,Def1;
        end;
        case
A8:      s = k * 2 + 1;
A9:      (2|^(n+1)) - 1 <= (2|^(n+1)) by XREAL_1:44;
          k * 2 <= (2|^(n+1)) - 1 by A3,A8,XREAL_1:19;
          then k * 2 <= (2|^(n+1)) by A9,XXREAL_0:2;
          then k * 2 <= (2|^n)* 2 by NEWTON:6;
          then
A10:      k <= ((2|^n)* 2)/2 by XREAL_1:77;
          take k;
          a = (k * 2 + 1-1)/(2|^(n+1)) by A2,A4,A8,Def5
            .= k * 2 /((2|^n)* 2) by NEWTON:6
            .= (k/(2|^n))*(2/2) by XCMPLX_1:76
            .= k/(2|^n);
          hence thesis by A10;
        end;
      end;
      hence thesis;
    end;
    hence thesis by A2,Def1;
  end;
  thus (axis(x)+1)/(2|^(n+1)) in dyadic(n)
  proof
    set a = (axis(x)+1)/(2|^(n+1));
    ex i being Nat st i <= (2|^n) & a = i/(2|^n)
    proof
      consider s being Nat such that
A11:  s <= (2|^(n+1)) and
A12:  x = s/(2|^(n+1)) by Def1;
      consider k being Element of NAT such that
A13:  s = 2 * k or s = 2 * k + 1 by SCHEME1:1;
      now
        per cases by A13;
        case
A14:      s = k * 2;
          then x = (k * 2)/((2|^n)* 2) by A12,NEWTON:6;
          then
A15:      x = k/(2|^n) by XCMPLX_1:91;
          k * 2 <= (2|^n)* 2 by A11,A14,NEWTON:6;
          then k <= ((2|^n)* 2)/2 by XREAL_1:77;
          hence thesis by A1,A15,Def1;
        end;
        case
A16:      s = k * 2 + 1;
          consider l being Nat such that
A17:      l = k + 1;
          s <> (2|^(n+1))
          proof
A18:        2|^(n+1) <> 0 by NEWTON:83;
            assume s = (2|^(n+1));
            then x = 1 by A12,A18,XCMPLX_1:60;
            hence thesis by A1,Th6;
          end;
          then (k * 2 + 1 + 1) + (-1) < (2|^(n+1)) by A11,A16,XXREAL_0:1;
          then l * 2 <= (2|^(n+1)) by A17,NAT_1:13;
          then l * 2 <= (2|^n)* 2 by NEWTON:6;
          then
A19:      l <= ((2|^n)* 2)/2 by XREAL_1:77;
          take l;
          a = (k * 2 + 1 + 1)/(2|^(n+1)) by A12,A16,Def5
            .= (k + 1) * 2 /((2|^n)* 2) by NEWTON:6
            .= ((k + 1)/(2|^n))*(2/2) by XCMPLX_1:76
            .= l/(2|^n) by A17;
          hence thesis by A19;
        end;
      end;
      hence thesis;
    end;
    hence thesis by Def1;
  end;
end;
