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
  q<>p & LSeg(q,p) /\ L~f = {q} implies not p in L~f
proof
  assume that
A1: q<>p and
A2: LSeg(q,p) /\ L~f = {q} & p in L~f;
  p in LSeg(q,p) by RLTOPSP1:68;
  then p in {q} by A2,XBOOLE_0:def 4;
  hence contradiction by A1,TARSKI:def 1;
end;
