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 Th16:
  not L1 // L2 implies
   (ex x st L1 = {x} or L2 = {x})
  or
   (L1 is being_line & L2 is being_line & ex x st L1 /\ L2 = {x})
  proof
    assume
A1: not L1 // L2;
    set n = 2;
    consider x1,x2 be Element of REAL 2 such that
A2: L1 = Line(x1,x2) by EUCLIDLP:51;
    consider y1,y2 be Element of REAL 2 such that
A3: L2 = Line(y1,y2) by EUCLIDLP:51;
    (x2-x1) = 0*n or (y2-y1) = 0*n or for r be Real holds
       not ((x2-x1) =r*(y2-y1)) by A1,A2,A3,EUCLIDLP:def 7,EUCLIDLP:def 1;
    then per cases by EUCLIDLP:9;
    suppose x1=x2 or y1=y2;
      then L1 = {x1} or L2 = {y1} by A2,A3,Th3;
      hence thesis;
    end;
    suppose
A4:   not(x1=x2) & (y1<>y2) & for r be Real holds (x2-x1) <> r*(y2-y1);
      x2-x1, y2-y1 are_lindependent2
      proof
        assume x2-x1, y2-y1 are_ldependent2;
        then x2-x1 = 0*n or y2-y1=0*n by A1,A2,A3,EUCLIDLP:def 7,EUCLIDLP:42;
        hence contradiction by A4,EUCLIDLP:9;
      end;
      then consider Pt be Element of REAL 2 such that
A5:   Pt in L1 and
A6:   Pt in L2 by A2,A3,Th13,EUCLIDLP:114,EUCLIDLP:49;
A7:   {Pt} c= L1 /\ L2
      proof
        let t be object;
        assume t in {Pt};
        then t = Pt by TARSKI:def 1;
        hence thesis by A5,A6,XBOOLE_0:def 4;
      end;
      L1 /\ L2 c= {Pt}
      proof
        let t be object;
        assume
A8:     t in L1 /\ L2;
        assume
        not t in {Pt};
        then
A9:     t <> Pt by TARSKI:def 1;
        reconsider t1=t as Element of REAL 2 by A8;
        t1 in L1 & t1 in L2 & Pt in L1 & Pt in L2 by A8,A5,A6,XBOOLE_0:def 4;
        then Line(t1,Pt) = L1 & Line(t1,Pt) = L2
             by A2,A3,A9,EUCLID_4:10,EUCLID_4:11;
        then L1 = L2 & L1 is being_line & L2 is being_line by A2,A4;
        hence contradiction by A1,EUCLIDLP:65;
      end;
      then L1 /\ L2 = {Pt} by A7;
      hence thesis by A4,A2,A3;
    end;
  end;
