reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem
  for l being quasi-loci holds varcl rng l = rng l
proof
  let l be quasi-loci;
  now
    let x,y;
    assume
A1: [x,y] in rng l;
    then reconsider xy = [x,y] as variable;
    consider i being object such that
A2: i in dom l and
A3: xy = l.i by A1,FUNCT_1:def 3;
    reconsider i as Nat by A2;
A4: vars xy = x;
    thus x c= rng l
    proof
      let a be object;
      assume
A5:   a in x;
      then reconsider a as variable by A4;
      ex j being Nat st j in dom l & j < i & a = l.j by A2,A3,A4,A5,Th30;
      hence thesis by FUNCT_1:def 3;
    end;
  end;
  hence varcl rng l c= rng l by Def1;
  thus thesis by Def1;
end;
