theorem Th1:
  for f being XFinSequence st n in dom XFS2FS(f) holds n-1 in dom f
  proof
    let f be XFinSequence;
    set g = XFS2FS(f);
    assume
A1: n in dom g;
    then reconsider n as non zero Nat by FINSEQ_3:25;
A2: len g = len f by AFINSQ_1:def 9;
    n <= len g by A1,FINSEQ_3:25;
    then n-1 < len f - 0 by A2,XREAL_1:8;
    hence thesis by AFINSQ_1:86;
  end;
