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

theorem Th13:
  for R being non degenerated Ring, S being Subring of R
  for f being Polynomial of S
  for g being non-zero Polynomial of R st f = g holds
  f is non-zero
  proof
    let R be non degenerated Ring;
    let S be Subring of R;
    let f be Polynomial of S;
    let g be non-zero Polynomial of R;
    assume f = g;
    then
A1: len f = len g by Th9;
    0 < len g by UPROOTS:17;
    hence thesis by A1,UPROOTS:17;
  end;
