reserve D,D1,D2 for non empty set,
        d,d1,d2 for XFinSequence of D,
        n,k,i,j for Nat;

theorem Th4:
  XFS2FS d1 = XFS2FS d2 implies d1 = d2
proof
  assume
A1: XFS2FS d1 = XFS2FS d2;
  set Xd1=XFS2FS d1,Xd2=XFS2FS d2;
A2: len Xd1 = len d1 by AFINSQ_1:def 9;
A3: len Xd2=len d2 by AFINSQ_1:def 9;
  for i st i < len d1 holds d1.i = d2.i
    proof
      let i such that
A4:     i < len d1;
A5:   i+1 <= len d1 by A4,NAT_1:13;
A6:   i+1-1=i;
      then d1.i = Xd1.(i+1) by NAT_1:11,A5,AFINSQ_1:def 9;
      hence thesis by A6,NAT_1:11,A5,A2,A3,A1,AFINSQ_1:def 9;
    end;
    hence thesis by A2,A3,A1,AFINSQ_1:9;
end;
