reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;
reserve
  A0 for (X,S)-terms non-empty MSAlgebra over S,
  A1 for all_vars_including (X,S)-terms MSAlgebra over S,
  A2 for all_vars_including inheriting_operations (X,S)-terms MSAlgebra over S,
  A for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S;
reserve X0 for non-empty countable ManySortedSet of S;
reserve A0 for all_vars_including inheriting_operations free_in_itself
  (X0,S)-terms MSAlgebra over S;

theorem Th81:
  for p being FinSequence holds p/^0 = p &
  for i being Nat st i >= len p holds p/^i = {}
  proof
    let p be FinSequence;
A1: 0 <= len p by NAT_1:2;
A2: now
      let i be Nat;
      assume 1 <= i & i <= len(p/^0);
      then i in dom(p/^0) by FINSEQ_3:25;
      hence (p/^0).i = p.(i+0) by A1,RFINSEQ:def 1 .= p.i;
    end;
    len(p/^0) = len p - 0 by A1,RFINSEQ:def 1 .= len p;
    hence p/^0 = p by A2;
    let i be Nat;
    assume i >= len p;
    then i = len p & len (p /^ len p) = len p - len p or i > len p
    by XXREAL_0:1,RFINSEQ:def 1;
    hence p/^i = {} by RFINSEQ:def 1;
  end;
