reserve n,i,k,m for Nat;
reserve r,r1,r2,s,s1,s2 for Real;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL 2,
  f,f1,f2 for FinSequence of the carrier of TOP-REAL 2,
  p,p1,p2,p3,q,q3 for Point of TOP-REAL 2;

theorem Th23:
  for f holds {LSeg(f,i): 1<=i & i+1<=len f}
     is finite
proof
  let f;
  set F = {LSeg(f,i): 1<=i & i+1<=len f},
F9 = {LSeg(f,i): 1<=i & i<=len f};
  F c= F9
  proof
    let x be object;
    assume x in F;
    then consider i being Nat such that
A1: x = LSeg(f,i) & 1<=i and
A2: i+1<=len f;
    i <= i + 1 by NAT_1:11;
    then i <= len f by A2,XXREAL_0:2;
    hence thesis by A1;
  end;
  hence thesis by Th22,FINSET_1:1;
end;
