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;
reserve F, F1 for FinSequence of INT,
  k, m, n, ma for Nat;
reserve i,j,k,m,n for Nat,
  D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve f for circular FinSequence of D;

theorem Th18:
  p in rng f & 1 <= i & i <= p..f implies
    f/.i = (Rotate(f,p))/.(i + len f -' p..f)
proof
  assume that
A1: p in rng f and
A2: 1 <= i and
A3: i <= p..f;
A4: len f -' p..f <= len f -' i by A3,NAT_D:41;
A5: p..f <= len f by A1,FINSEQ_4:21;
  then i <= len f by A3,XXREAL_0:2;
  then len f -' p..f + i <= len f by A4,NAT_D:54;
  then
A6: i + len f -' p..f <= len f by A5,NAT_D:38;
  len f + 1 <= i + len f by A2,XREAL_1:6;
  then len f + 1 - p..f <= i + len f - p..f by XREAL_1:9;
  then len f - p..f + 1 <= i + (len f - p..f);
  then len f - p..f + 1 <= i + (len f -' p..f) by A5,XREAL_1:233;
  then len f - p..f + 1 <= i + len f -' p..f by A5,NAT_D:38;
  then
A7: len(f:-p) <= i + len f -' p..f by A1,FINSEQ_5:50;
  len f <= i + len f by NAT_1:11;
  then i + len f -' p..f + p..f -' len f = i + len f -' len f
    by A5,XREAL_1:235,XXREAL_0:2
    .= i by NAT_D:34;
  hence thesis by A1,A7,A6,Th17;
end;
