reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;
reserve lambda for Real;

theorem
  f is being_S-Seq & f/.len f in LSeg(f,m) & 1<=m & m + 1 <=len f
  implies m + 1 = len f
proof
  assume that
A1: f is being_S-Seq and
A2: f/.len f in LSeg(f,m) and
A3: 1<=m and
A4: m + 1 <= len f;
A5: f is s.n.c. by A1;
A6: f is one-to-one by A1;
A7: f is unfolded by A1;
  set q= f/.len f;
A8: m+1+1=m+(1+1);
A9: len f >=2 by A1;
  then reconsider k = len f - 1 as Element of NAT by INT_1:5,XXREAL_0:2;
A10: k+1 = len f;
  assume m + 1 <> len f;
  then
A11: m+1<=k by A4,A10,NAT_1:8;
  1<=len f by A9,XXREAL_0:2;
  then
A12: len f in dom f by FINSEQ_3:25;
  per cases by A11,XXREAL_0:1;
  suppose
A13: m+1=k;
A14: 1<=m+1 by NAT_1:11;
    m+1+1<=len f by A13;
    then
A15: f/.(m+2) in LSeg(f,m+1) by A14,TOPREAL1:21;
    LSeg(f,m) /\ LSeg(f,m+1) = {f/.(m+1)} by A3,A7,A8,A13;
    then q in {f/.(m+1)} by A2,A13,A15,XBOOLE_0:def 4;
    then
A16: f/.len f=f/.(m+1) by TARSKI:def 1;
    m+1<=len f by A10,A13,NAT_1:11;
    then m+1 in dom f by A14,FINSEQ_3:25;
    then len f= len f -1 by A6,A12,A13,A16,PARTFUN2:10;
    hence contradiction;
  end;
  suppose
A17: m+1<k;
    (1+1)-1 = 1;
    then k+1 = len f & 1<=k by A9,XREAL_1:13;
    then
A18: q in LSeg(f,k) by TOPREAL1:21;
    LSeg(f,m) misses LSeg(f,k) by A5,A17;
    then LSeg(f,m) /\ LSeg(f,k) = {};
    hence contradiction by A2,A18,XBOOLE_0:def 4;
  end;
end;
