
theorem lemman02: :: NEWTON02:146 generalized for integers
  for a,b be Integer, m be positive Nat holds
  Sum (a,b) In_Power m = a|^m + b|^m + Sum ((((a,b) In_Power m)|m)/^1)
  proof
    let a,b be Integer;
    let m be positive Nat;
    len ((a,b) In_Power m) = m + 1 by NEWTON:def 4; then
    Sum ((a,b) In_Power m) = Sum(((a,b) In_Power m)|m/^1) +
    ((a,b) In_Power m).1 + ((a,b) In_Power m).(m+1) by NEWTON02:115
    .= Sum(((a,b) In_Power m)|m/^1) + a|^m + ((a,b) In_Power m).(m+1)
    by NEWTON:28
    .= Sum(((a,b) In_Power m)|m/^1) + a|^m + b|^m by NEWTON:29;
    hence thesis;
  end;
