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
  Sum((a,b) In_Power (m+n)) = Sum((a,b) In_Power m)*Sum((a,b) In_Power n)
  proof
    Sum((a,b) In_Power (m+n)) = (a+b)|^(m+n) by NEWTON:30
    .=(a+b)|^m*(a+b)|^n by NEWTON:8
    .= Sum((a,b) In_Power m)*(a+b)|^n by NEWTON:30
    .= Sum((a,b) In_Power m)*Sum((a,b) In_Power n) by NEWTON:30;
    hence thesis;
  end;
