reserve a,b,i,j,k,l,m,n for Nat;

theorem FINSEQ74:
  for f be FinSequence, i be non zero Nat holds f.(n+i) = (f/^n).i
  proof
    let f be FinSequence, i be non zero Nat;
    consider D be non empty set such that
    A0: f is FinSequence of D by Th0;
    reconsider f as FinSequence of D by A0;
    per cases;
    suppose
      B1: not i in dom (f/^n);
      B2: (f/^n).i = 0 by B1,FUNCT_1:def 2;
      n+i > len f
      proof
        i >= 1 by NAT_1:14; then
        i > len (f/^n) by B1, FINSEQ_3:25; then
        C1: i > len f -' n by RFINSEQ:29;
        per cases;
        suppose
          D1: len f < n;
          n+i >= n+0 by XREAL_1:6;
          hence thesis by D1,XXREAL_0:2;
        end;
        suppose
          len f >= n; then
          len f - n >= n - n by XREAL_1:9; then
          len f -' n = len f - n by XREAL_0:def 2; then
          i+n > (len f - n)+n by C1,XREAL_1:6;
          hence thesis;
        end;
      end; then
      not n+i in dom f by FINSEQ_3:25;
      hence thesis by B2,FUNCT_1:def 2;
    end;
    suppose
      i in dom (f/^n);
      hence thesis by FINSEQ_7:4;
    end;
  end;
