reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

theorem
  ex L st L is being_point & L misses L1
  proof
    consider x such that
A1: not x in L1 by Th15;
    x in REAL 2;
    then {x} c= REAL 2 by TARSKI:def 1;
    then reconsider L = {x} as Subset of REAL 2;
    L = Line(x,x) by Th3;
    then reconsider L as Element of line_of_REAL 2 by EUCLIDLP:47;
A2: L is being_point;
    now
      assume L meets L1;
      then consider y be object such that
A3:   y in L and
A4:   y in L1 by XBOOLE_0:3;
      thus contradiction by A1,A3,A4,TARSKI:def 1;
    end;
    hence thesis by A2;
  end;
