theorem
  x in rng p implies (x..p = len p iff p |-- x = {})
proof
  assume
A1: x in rng p;
  thus x..p = len p implies p |-- x = {}
  proof
    assume
A2: x..p = len p;
    len(p |-- x) = len p - x..p by A1,Def6
      .= 0 by A2;
    hence thesis;
  end;
  assume p |-- x = {};
  then
A3: len(p |-- x) = 0;
  len(p |-- x) = len p - x..p by A1,Def6;
  hence thesis by A3;
end;
