reserve n,k,b for Nat, i for Integer;

theorem Th11:
  for F being XFinSequence
  holds F = SubXFinS(F,NAT)
  proof
    let F be XFinSequence;
    set SFN=Sgm0(len F /\ NAT);
    dom F c= NAT;
    then A1: len F /\ NAT = len F by XBOOLE_1:28;
    A2: F is Function of dom F,rng F by FUNCT_2:1;
    reconsider idF= id dom F as XFinSequence of NAT;
    A3: rng idF = len F;
    now
      let l,m,k1,k2 be Nat;
      assume A4: l < m & m < len idF & k1=idF.l & k2=idF.m;
      then l < len idF by XXREAL_0:2;
      then k1=l & k2=m by A4,FUNCT_1:18,AFINSQ_1:66;
      hence k1 < k2 by A4;
    end;
    then id dom F = Sgm0(Segm len F) by AFINSQ_2:def 4,A3;
    hence thesis by A1,FUNCT_2:17,A2;
  end;
