reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem Th13:
  for f be FinSequence of TOP-REAL 2 for p be Point of TOP-REAL 2
  for i1 be Element of NAT st f is s.n.c. & p in LSeg(f,i1) holds i1=Index(p,f)
  or i1=Index(p,f)+1
proof
  let f be FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  let i1 be Element of NAT;
  assume that
A1: f is s.n.c. and
A2: p in LSeg(f,i1);
  p in L~f by A2,SPPOL_2:17;
  then p in LSeg(f,Index(p,f)) by Th9;
  then p in LSeg(f,Index(p,f)) /\ LSeg(f,i1) by A2,XBOOLE_0:def 4;
  then
A3: LSeg(f,Index(p,f)) meets LSeg(f,i1) by XBOOLE_0:4;
  assume
A4: not thesis;
  Index(p,f) <= i1 by A2,Th7;
  then Index(p,f) < i1 by A4,XXREAL_0:1;
  then Index(p,f)+1 <= i1 by NAT_1:13;
  then Index(p,f)+1 < i1 by A4,XXREAL_0:1;
  hence contradiction by A1,A3,TOPREAL1:def 7;
end;
