reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;

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;
