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 Th12:
  for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL
2,i1 being Nat st f is being_S-Seq & 1<i1 & i1<=len f & p=f.i1 holds Index(p,f)
  + 1 = i1
proof
  let f be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2,i1 be Nat;
  assume
A1: f is being_S-Seq;
  assume that
A2: 1<i1 and
A3: i1<=len f;
A4: i1 in dom f by A2,A3,FINSEQ_3:25;
  assume p=f.i1;
  then
A5: p = f/.i1 by A4,PARTFUN1:def 6;
  assume
A6: Index(p,f) + 1 <> i1;
  consider j being Nat such that
A7: i1 = j+1 by A2,NAT_1:6;
  reconsider j as Element of NAT by ORDINAL1:def 12;
A8: 1 + 0 <= j by A2,A7,NAT_1:13;
  then
A9: p in LSeg(f,j) by A3,A7,A5,TOPREAL1:21;
  then Index(p,f) <= j by Th7;
  then Index(p,f) < j by A7,A6,XXREAL_0:1;
  then
A10: Index(p,f) + 1 <= j by NAT_1:13;
A11: LSeg(f,j) c= L~f by TOPREAL3:19;
  then
A12: p in LSeg(f,Index(p,f)) by A9,Th9;
  per cases by A10,XXREAL_0:1;
  suppose
A13: Index(p,f) + 1 = j;
    then
A14: Index(p,f) + (1+1) <= len f by A3,A7;
    1 <= Index(p,f) by A9,A11,Th8;
    then LSeg(f,Index(p,f)) /\ LSeg(f,j) = {f/.j} by A1,A13,A14,TOPREAL1:def 6;
    then p in {f/.j} by A9,A12,XBOOLE_0:def 4;
    then
A15: p = f/.j by TARSKI:def 1;
    j < len f by A3,A7,NAT_1:13;
    then
A16: j in dom f by A8,FINSEQ_3:25;
    j < i1 by A7,NAT_1:13;
    hence contradiction by A1,A4,A5,A15,A16,PARTFUN2:10;
  end;
  suppose
A17: Index(p,f) + 1 < j;
    p in LSeg(f,Index(p,f)) /\ LSeg(f,j) by A9,A12,XBOOLE_0:def 4;
    then LSeg(f,Index(p,f)) meets LSeg(f,j) by XBOOLE_0:4;
    hence contradiction by A1,A17,TOPREAL1:def 7;
  end;
end;
