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 Th14:
  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 unfolded s.n.c. & i1+1<=len f & p in LSeg(f,i1
  ) & p <> f.i1 holds i1=Index(p,f)
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 unfolded s.n.c. and
A2: i1+1<=len f and
A3: p in LSeg(f,i1);
A4: i1 < len f by A2,NAT_1:13;
A5: 1 <= Index(p,f)+1 by NAT_1:11;
  Index(p,f) <= i1 by A3,Th7;
  then Index(p,f) < len f by A4,XXREAL_0:2;
  then Index(p,f)+1 <= len f by NAT_1:13;
  then
A6: Index(p,f) + 1 in dom f by A5,FINSEQ_3:25;
  assume
A7: p <> f.i1;
A8: p in L~f by A3,SPPOL_2:17;
  then p in LSeg(f,Index(p,f)) by Th9;
  then
A9: p in LSeg(f,Index(p,f)) /\ LSeg(f,i1) by A3,XBOOLE_0:def 4;
A10: 1 <= Index(p,f) by A8,Th8;
  now
    assume
A11: i1 = Index(p,f)+1;
    then Index(p,f) + (1+1) <= len f by A2;
    then p in {f/.(Index(p,f)+1)} by A1,A9,A10,A11,TOPREAL1:def 6;
    then p = f/.(Index(p,f)+1) by TARSKI:def 1;
    hence contradiction by A7,A6,A11,PARTFUN1:def 6;
  end;
  hence thesis by A1,A3,Th13;
end;
