reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem
  not 3 divides n iff ex k st n = 3*k+1 or n = 3*k+2
  proof
    consider K being Nat such that
A1: n = 3*K or n = 3*K+1 or n = 3*K+2 by Th23;
    thus not 3 divides n implies ex k st n = 3*k+1 or n = 3*k+2 by A1;
    given k such that
A2: n = 3*k+1 or n = 3*k+2;
    given t being Nat such that
A3: n = 3*t;
    per cases by A2;
    suppose n = 3*k+1;
      then 1/3 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 3*k+2;
      then 2/3 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
  end;
