
theorem Th41:
  for L being non degenerated non empty multLoopStr_0, n being
Element of NAT, p being Polynomial of L st n <= len p holds len poly_shift(p,n)
  + n = len p
proof
  let L be non degenerated non empty multLoopStr_0, n be Element of NAT, p
  be Polynomial of L such that
A1: n <= len p;
  set ps = poly_shift(p,n), lpn = len p - n;
  n-n <= lpn by A1,XREAL_1:9;
  then reconsider lpn as Element of NAT by INT_1:3;
A2: now
    let m be Nat such that
A3: m is_at_least_length_of ps and
A4: lpn > m;
    lpn >= m+1 by A4,NAT_1:13;
    then
A5: lpn -1 >= m +1-1 by XREAL_1:9;
    then reconsider lpn1 = lpn -1 as Element of NAT by INT_1:3;
    (n+lpn1)+1 = len p;
    then
A6: p.(n+lpn1) <> 0.L by ALGSEQ_1:10;
    ps.lpn1 = p.(n+lpn1) by Def5;
    hence contradiction by A3,A5,A6,ALGSEQ_1:def 2;
  end;
  now
    let i be Nat;
    assume i >= lpn;
    then
A7: i+n >= len p by XREAL_1:20;
    thus ps.i = p.(n+i) by Def5
      .= 0.L by A7,ALGSEQ_1:8;
  end;
  then lpn is_at_least_length_of ps by ALGSEQ_1:def 2;
  hence len poly_shift(p,n) + n = lpn + n by A2,ALGSEQ_1:def 3
    .= len p;
end;
