theorem Th31:
  p+q divides u-v implies p+q divides p*(u+t) + q*(v+t)
  proof
    set z = u-v;
    assume p+q divides u-v; then
    p+q divides p*z + q*0 by INT_2:2;
    then p+q divides p*(z+(v+t)) + q*(0+(v+t)) by Th29;
    hence thesis;
  end;
