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

theorem Th29:
  p+q divides p*u + q*v implies p+q divides p*(u+z) + q*(v+z)
  proof
    assume p+q divides p*u + q*v; then
    consider t be Integer such that
    A2: (p+q)*t = p*u + q*v;
    p*(u+z) + q*(v+z) = (p+q)*(t+z) by A2;
    hence thesis;
  end;
