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;
reserve f,g for FinSequence of TOP-REAL 2;
reserve p for Point of TOP-REAL 2,
  f for FinSequence of TOP-REAL 2;
reserve f for circular FinSequence of TOP-REAL 2;

theorem Th32:
  p in rng f & 1 <= i & i < p..f implies LSeg(f,i) = LSeg(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: p..f <= len f by A1,FINSEQ_4:21;
A5: i + len f < len f + p..f by A3,XREAL_1:6;
A6: len f <= i + len f by NAT_1:11;
  then p.. f <= i + len f by A4,XXREAL_0:2;
  then
A7: i + len f -' p..f < len f by A5,NAT_D:54;
  len f + 1 <= i + len f by A2,XREAL_1:6;
  then len f + 1 -' p..f <= i + len f -' p..f by NAT_D:42;
  then len f -' p..f + 1 <= i + len f -' p..f by A4,NAT_D:38;
  then len f - p..f + 1 <= i + len f -' p..f by A4,XREAL_1:233;
  then
A8: len(f:-p) <= i + len f -' p..f by A1,FINSEQ_5:50;
  i + len f -' p..f + p..f -' len f = i + len f -' len f by A4,A6,XREAL_1:235
,XXREAL_0:2
    .= i by NAT_D:34;
  hence thesis by A1,A8,A7,Th31;
end;
