
theorem lemcontr:
not ex r being Rational st r|^3 = 2
proof
now assume ex r being Rational st r|^3 = 2; then
   consider r being Rational such that A1: r|^3 = 2;
   consider i being Integer, n being Nat such that
A3: n <> 0 and
A4: r = i / n and
A5: for i1 being Integer, n1 being Nat
    st n1 <> 0 & r = i1 / n1 holds n <= n1 by RAT_1:9;
A6: i = r * n by A3,A4,XCMPLX_1:87;
    r >= 0 by A1,lemcontr1;
  then reconsider m = i as Element of NAT by A6,INT_1:3;
A7: m|^3 = 2 * n|^3 by A1,A6,lemcontr2;
  then m is even;
  then consider m1 being Nat such that
A8: m = 2 * m1 by ABIAN:def 2;
  n|^3 = ((2 * m1)|^3) / 2 by A7,A8
      .= ((2 |^ (2+1)) * (m1 |^ 3)) / 2 by lemcontr2
      .= (((2|^2 * 2) * (m1 |^ 3))) / 2 by NEWTON:6
      .= 2|^(1+1) * m1|^3
      .= (2|^1 * 2) * m1|^3 by NEWTON:6
      .= 2 * (2|^1 * m1|^3);
  then n is even;
  then consider n1 being Nat such that
A9: n = 2 * n1 by ABIAN:def 2;
  reconsider n1 as Element of NAT by ORDINAL1:def 12;
A10: m1/n1 = r by A4,A8,A9,XCMPLX_1:91;
A11: n1 <> 0 by A3,A9;
  then 2 * n1 > 1 * n1 by XREAL_1:98;
  hence contradiction by A5,A9,A11,A10;
  end;
hence thesis;
end;
