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 Th15:
  for i,j be Nat holds f = <* p,p1,q *> & i<>0 & j>i+1 implies LSeg(f,j) = {}
proof
  let i,j be Nat;
  assume that
A1: f = <* p,p1,q *> and
A2: i<>0 and
A3: j>i+1;
  i>=0+1 by A2,NAT_1:13;
  then 1+i>=1+1 by XREAL_1:7;
  then j>2 by A3,XXREAL_0:2;
  then j>=2+1 by NAT_1:13;
  then
A4: j+1 > 3 by NAT_1:13;
  len f = 3 by A1,FINSEQ_1:45;
  hence thesis by A4,TOPREAL1:def 3;
end;
