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

theorem Th30:
  for l being one-to-one FinSequence of Vars holds l is quasi-loci iff
  for i being Nat, x being variable st i in dom l & x = l.i
  for y being variable st y in vars x
  ex j being Nat st j in dom l & j < i & y = l.j
proof
  let l be one-to-one FinSequence of Vars;
  thus
  now
    assume
A1: l is quasi-loci;
    let i be Nat, x be variable such that
A2: i in dom l and
A3: x = l.i;
    let y be variable such that
A4: y in vars x;
    vars x c= rng (l|(i qua set)) by A1,A2,A3,Def3;
    then consider z being object such that
A5: z in dom (l dom i) and
A6: y = (l dom i).z by A4,FUNCT_1:def 3;
A7: dom (l dom i) = dom l /\ i by RELAT_1:61;
    reconsider z as Element of NAT by A5,A7;
    reconsider j = z as Nat;
    take j;
A8: card Segm z = z;
    card Segm i = i;
    hence j in dom l & j < i & y = l.j by A5,A6,A7,A8,FUNCT_1:47,NAT_1:41
,XBOOLE_0:def 4;
  end;
  assume
A9: for i being Nat, x being variable st i in dom l & x = l.i
  for y being variable st y in vars x
  ex j being Nat st j in dom l & j < i & y = l.j;
  now
    let i;
    assume
A10: i in dom l;
    then l.i in rng l by FUNCT_1:def 3;
    then reconsider x = l.i as variable;
    thus (l.i)`1 c= rng (l dom i)
    proof
      let y be object;
      assume y in (l.i)`1;
      then
A11:  y in vars x;
      then reconsider y as variable;
      consider j being Nat such that
A12:  j in dom l and
A13:  j < i and
A14:  y = l.j by A9,A10,A11;
A15:  card Segm i = i;
      card Segm j = j;
      then j in i by A13,A15,NAT_1:41;
      hence thesis by A12,A14,FUNCT_1:50;
    end;
  end;
  hence thesis by Def3;
end;
