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
  len f>=2 & not p in L~f implies for n st 1<=n & n<=len f holds f/.n<>p
proof
  assume that
A1: len f>=2 and
A2: not p in L~f;
  set Mf={LSeg(f,k): 1<=k & k+1<=len f};
  given n such that
A3: 1<=n and
A4: n<=len f and
A5: f/.n=p;
  per cases by A4,XXREAL_0:1;
  suppose
A6: n=len f;
    reconsider j = len f - 1 as Element of NAT by A1,INT_1:5,XXREAL_0:2;
    1+1<=len f by A1;
    then
A7: 1<=j by XREAL_1:19;
    then
A8: f/.(j+1) in LSeg(f,j) by TOPREAL1:21;
    j+1<=len f;
    then LSeg(f,j) in Mf by A7;
    hence contradiction by A2,A5,A6,A8,TARSKI:def 4;
  end;
  suppose
A9: n<len f;
    then n+1<=len f by NAT_1:13;
    then
A10: f/.n in LSeg(f,n) by A3,TOPREAL1:21;
    n+1<=len f by A9,NAT_1:13;
    then LSeg(f,n) in Mf by A3;
    hence contradiction by A2,A5,A10,TARSKI:def 4;
  end;
end;
