reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;
reserve D for non empty set;
reserve p, q for FinSequence,
  X, Y, x, y for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;
reserve a, a1, a2 for TwoValued Alternating FinSequence;
reserve fs, fs1, fs2 for FinSequence of X,
  fss, fss2 for Subset of fs;

theorem :: GRAPH_2:15
  for f1, f2 being FinSequence of D holds 1<=n & n<len f2 implies
  (f1^'f2)/.(len f1 + n) = f2/.(n+1)
proof
  let f1, f2 be FinSequence of D;
  assume that
A1: 1<=n and
A2: n<len f2;
A3: n+1 <= len f2 by A2,NAT_1:13;
A4: now
    per cases;
    suppose
A5:   f2 <> {};
A6:   len f1 + n < len f1 + len f2 by A2,XREAL_1:6;
      len (f1^'f2) + 1 = len f1 + len f2 by A5,Th13;
      hence len f1+n <= len (f1^'f2) by A6,NAT_1:13;
    end;
    suppose
      f2 = {};
      hence len f1+n <= len (f1^'f2) by A2;
    end;
  end;
A7: 0+1 <= n+1 by XREAL_1:6;
  0+1 <= len f1 + n by A1,XREAL_1:7;
  hence (f1^'f2)/.(len f1 + n) = (f1^'f2).(len f1 + n) by A4,FINSEQ_4:15
    .= f2.(n+1) by A1,A2,Th15
    .= f2/.(n+1) by A7,A3,FINSEQ_4:15;
end;
