
theorem
  for i being Nat, f being FinSequence of TOP-REAL 2,
  P being Subset of TOP-REAL 2 st
  f is being_S-Seq & 1 <= i & i+1 <= len f & P = LSeg(f,i) holds
  P is_an_arc_of f/.i, f/.(i+1)
proof
  let i be Nat, f be FinSequence of TOP-REAL 2,
  P be Subset of TOP-REAL 2;
  assume that
A1: f is being_S-Seq and
A2: 1 <= i and
A3: i+1 <= len f and
A4: P = LSeg(f,i);
A5: i in dom f by A2,A3,SEQ_4:134;
A6: i+1 in dom f by A2,A3,SEQ_4:134;
A7: f is one-to-one by A1;
A8: LSeg (f,i) = LSeg (f/.i, f/.(i+1)) by A2,A3,TOPREAL1:def 3;
  f/.i <> f/.(i+1)
  proof
    assume f/.i = f/.(i+1);
    then i = i + 1 by A5,A6,A7,PARTFUN2:10;
    hence thesis;
  end;
  hence thesis by A4,A8,TOPREAL1:9;
end;
