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 Th48:
  for 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 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 Th17
  .= 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;
