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 Th22:
  for f holds {LSeg(f,i): 1<=i & i<=len f} is finite
proof
  defpred X[set] means not contradiction;
  deffunc U(FinSequence of TOP-REAL 2, Nat) = LSeg($1,$2);
  let f;
  set Y = {LSeg(f,i): 1<=i & i<=len f};
  set X = {U(f,i): 1<=i & i<=len f & X[i]};
A1: for e being object holds e in X iff e in Y
  proof
    let e be object;
    thus e in X implies e in Y
    proof
      assume e in X;
      then ex i being Nat st e = U(f,i) & 1<=i & i<=len f & X[i];
      hence thesis;
    end;
    assume e in Y;
    then ex i being Nat st e = LSeg(f,i) & 1<=i & i<=len f;
    hence thesis;
  end;
  X is finite from FinSeqFam9;
  hence thesis by A1,TARSKI:2;
end;
