reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th50:
  not 3 divides a*b implies 3 divides (a+b)*(a-b)
  proof
    assume not 3 divides a*b; then
    not 3 gcd a = |.3.| & not 3 gcd b = |.3.| by INT_2:2,Th3; then
    A3: 3,a|^2 are_coprime & 3,b|^2 are_coprime by PEPIN:2,41,WSIERP_1:10; then
    3 gcd a|^2 <> |.3.|; then
    not a|^2-0,0 are_congruent_mod 3 by Th3; then
    A4: a|^2, 1 are_congruent_mod 3 by NAT_6:15;
    3 gcd b|^2 <> |.3.| by A3; then
    not b|^2-0,0 are_congruent_mod 3 by Th3; then
    b|^2, 1 are_congruent_mod 3 by NAT_6:15; then
    a|^2 - b|^2, 1-1 are_congruent_mod 3 by A4,INT_1:17;
    hence thesis by NEWTON01:1;
  end;
