reserve m,n for Nat;
reserve r for Real;
reserve c for Element of F_Complex;

theorem Th8:
  for R being Ring, S being Subring of R
  for f being Polynomial of S holds
  f is Polynomial of R
  proof
    let R be Ring, S be Subring of R;
    let f be Polynomial of S;
    the carrier of S c= the carrier of R by C0SP1:def 3;
    then reconsider f as sequence of R by FUNCT_2:7;
    f is finite-Support
    proof
      consider n such that
A1:   for i being Nat st i >= n holds f.i = 0.S by ALGSEQ_1:def 1;
      take n;
      0.S = 0.R by C0SP1:def 3;
      hence thesis by A1;
    end;
    hence thesis;
  end;
