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 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;
