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

theorem Th9:
  for R being Ring, S being Subring of R
  for f being Polynomial of S
  for g being Polynomial of R st f = g
  holds len f = len g
  proof
    let R be Ring;
    let S be Subring of R;
    let f be Polynomial of S;
    let g be Polynomial of R;
    assume
A1: f = g;
A2: len g is_at_least_length_of f
    proof
      let i be Nat;
      assume i >= len g;
      then g.i = 0.R by ALGSEQ_1:8;
      hence f.i = 0.S by A1,C0SP1:def 3;
    end;
    for m being Nat st m is_at_least_length_of f holds len g <= m
    proof
      let m be Nat;
      assume
A3:   m is_at_least_length_of f;
      m is_at_least_length_of g
      proof
        let i be Nat;
        assume i >= m;
        then f.i = 0.S by A3;
        hence g.i = 0.R by A1,C0SP1:def 3;
      end;
      hence len g <= m by ALGSEQ_1:def 3;
    end;
    hence thesis by A2,ALGSEQ_1:def 3;
  end;
