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;

theorem Th10:
  p in rng f & p..f <= i & i <= len f implies
    f/.i = (Rotate(f,p))/.(i+1 -' p..f)
proof
  assume that
A1: p in rng f and
A2: p..f <= i and
A3: i <= len f;
A4: 1 + p..f <= i+1 by A2,XREAL_1:6;
  i+1 <= len f + 1 by A3,XREAL_1:6;
  then i <= i+1 & i+1 - p..f <= len f + 1 - p..f by NAT_1:11,XREAL_1:9;
  then i+1 -' p..f <= len f - p..f + 1 by A2,XREAL_1:233,XXREAL_0:2;
  then
A5: i+1 -' p..f <= len(f:-p) by A1,FINSEQ_5:50;
  i+1 -' p..f -' 1 + p..f = i -' p..f+1 -' 1 + p..f by A2,NAT_D:38
    .= i -' p..f + p..f by NAT_D:34
    .= i by A2,XREAL_1:235;
  hence thesis by A1,A4,A5,Th9,NAT_D:55;
end;
