reserve a,b,i,j,k,l,m,n for Nat;

theorem DOMN:
for a,b be Real, n be Nat holds
len ((a,b) In_Power n) = len (Newton_Coeff n) &
len ((a,b) Subnomial n) = len (Newton_Coeff n) &
len ((a,b) In_Power n) = len ((a,b) Subnomial n) &
dom ((a,b) In_Power n) = dom (Newton_Coeff n) &
dom ((a,b) Subnomial n) = dom (Newton_Coeff n) &
dom ((a,b) In_Power n) = dom ((a,b) Subnomial n)
proof
  let a,b be Real, n be Nat;
  len (a,b) In_Power (n+1-1) = len (Newton_Coeff n); then
  A2: dom ((a,b) In_Power n) = dom (Newton_Coeff n) by FINSEQ_3:29;
  dom (((a,b) In_Power n) /" (Newton_Coeff n)) =
  dom ((a,b) In_Power n) /\ dom (Newton_Coeff n) by VALUED_1:16;
  hence thesis by A2,FINSEQ_3:29;
end;
