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;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem 3 divides a*b or 3 divides a+b or 3 divides a-b
  proof
    per cases;
    suppose 3 divides a or 3 divides b;
      hence thesis by INT_2:2;
    end;
    suppose not 3 divides a & not 3 divides b; then
      per cases by Th90;
      suppose
        3 divides a+1 & 3 divides b+1; then
        3 divides (a+1)-(b+1) by INT_5:1;
        hence thesis;
      end;
      suppose
        3 divides a - 1 & 3 divides b - 1; then
        3 divides (a-1) - (b-1) by INT_5:1;
        hence thesis;
      end;
      suppose
        3 divides a - 1 & 3 divides b + 1;then
        3 divides (a-1)+(b+1) by WSIERP_1:4;
        hence thesis;
      end;
      suppose
        3 divides a + 1 & 3 divides b - 1;then
        3 divides (a+1)+ (b-1) by WSIERP_1:4;
        hence thesis;
      end;
    end;
  end;
