reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem ODD:
  for t be Integer holds t is odd iff t gcd 2 = 1
  proof
    let t be Integer;
    thus t is odd implies t gcd 2 = 1
    proof
      assume t is odd; then
      consider z be Integer such that
  A1: t = 2*z + 1 by ABIAN:1;
      t gcd 2 = 1 gcd (1 + 1*1) by A1,NEWTON02:5
      .= 1;
      hence thesis;
    end;
    t gcd 2 <> |.2.| implies not 2 divides t by NEWTON02:3;
    hence thesis;
  end;
