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
  (k+l) is prime & k > 0 & l > 0 implies
    (k+l) divides ((a,b) In_Power (k+l)).(k+1)
  proof
    assume
    A1: (k+l) is prime & k > 0 & l > 0; then
    k+l > k+0 by XREAL_1:6; then
    A2: (k+l) divides ((k+l) choose k) by A1,Th21;
    ((k+l)choose k) divides ((k+l) choose k)*((a|^l)*(b|^k)); then
    consider x such that
    A3: ((k+l) choose k)*((a|^l)*(b|^k)) = (k+l)*x by A2,INT_2:9,NAT_D:def 3;
    ((a,b) In_Power (k+l)).(k+1) = ((k+l) choose k)*(a|^l)*(b|^k) by Lm1
    .=(k+l)*x by A3;
    hence thesis;
  end;
