reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;

theorem
  r <> 0 & j <= i implies r|^(i-'j) = r|^i / r|^j
proof
  assume that
A1: r <> 0 and
A2: j <= i;
   reconsider rr=r as Real;
  thus r|^i / r|^j = (Product(i |-> rr)) / rr|^j by NEWTON:def 1
    .= (Product(i |-> rr)) / (Product(j |-> rr)) by NEWTON:def 1
    .= Product((i-'j) |-> rr) by A1,A2,Th8
    .= r|^(i-'j) by NEWTON:def 1;
end;
