reserve p,q,r for FinSequence,
  x,y for object;

theorem
  for p,q being FinSequence st p <> {} & q <> {}
  holds len(p$^q) + 1 = len p + len q
proof
  let p,q be FinSequence;
  assume p <> {} & q <> {};
  then consider i being Nat, r being FinSequence such that
    A1: len p = i + 1 & r = p|Seg i & p$^q = r^q by Def1;
  A2: i <= len p by A1, NAT_1:11;
  thus len(p$^q) + 1 = len r + len q + 1 by A1, FINSEQ_1:22
    .= i + len q + 1 by A1, A2, FINSEQ_1:17
    .= len p + len q by A1;
end;
