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;

theorem Th25:
  p in rng f & p..f <= i & i < len f implies LSeg(f,i) = LSeg(
  Rotate(f,p),i -' p..f+1)
proof
  assume that
A1: p in rng f and
A2: p..f <= i and
A3: i < len f;
  i - p..f < len f - p..f by A3,XREAL_1:9;
  then i -' p..f < len f - p..f by A2,XREAL_1:233;
  then i -' p..f+1 < len f - p..f + 1 by XREAL_1:6;
  then
A4: i -' p..f+1 < len(f:-p) by A1,FINSEQ_5:50;
  1 + p..f <= i + 1 by A2,XREAL_1:6;
  then 1 <= i + 1 -' p..f by NAT_D:55;
  then
A5: 1 <= i -' p..f+1 by A2,NAT_D:38;
  i -' p..f+1 -' 1 + p..f =i -' p..f + p..f by NAT_D:34
    .= i by A2,XREAL_1:235;
  hence thesis by A1,A5,A4,Th24;
end;
