reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;
reserve l for Nat;
reserve M for Nat;
reserve m,n for Nat;
reserve x1,x2,x3,x4 for object;
reserve e,u for object;

theorem Th13:
  for D being set, p being FinSequence of D, n being Nat
  holds n+1 in dom p iff n in dom FS2XFS p
proof
  let D be set, p be FinSequence of D, n be Nat;
  hereby
    assume n+1 in dom p;
    then n+1 <= len p by FINSEQ_3:25;
    then n+1-1 < len p-0 by XREAL_1:15;
    then n < len FS2XFS p by Def8;
    then n in Segm dom FS2XFS p by NAT_1:44;
    hence n in dom FS2XFS p;
  end;
  assume n in dom FS2XFS p;
  then n in Segm dom FS2XFS p;
  then 0 <= n & n < len FS2XFS p by NAT_1:44;
  then 0+1 <= n+1 & n < len p by Def8, XREAL_1:6;
  then 1 <= n+1 & n+1 <= len p by NAT_1:13;
  hence n+1 in dom p by FINSEQ_3:25;
end;
