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
  L1 _|_ L2 & C in L1 /\ L2 & A in L1 & B in L2 & A <> C & B <> C
  implies A,B,C is_a_triangle
  proof
    assume that
A1: L1 _|_ L2 and
A2: C in L1 /\ L2 and
A3: A in L1 and
A4: B in L2  and
A5: A<>C and
A6: B<>C;
    not A in Line(B,C)
    proof
      assume
A7:   A in Line(B,C);
A8:   C in L1 & C in L2 by A2,XBOOLE_0:def 4;
A9:   L1 is being_line & L2 is being_line by A1,EUCLIDLP:67;
      consider x such that
A10:  L1 /\ L2 = {x} by A1,Th33;
A11:  L1/\L2 = {C} by A2,A10,TARSKI:def 1;
      A in L2 & A in L1 by A3,A7,A4,A9,A8,A6,Th37;
      then A in {C} by A11,XBOOLE_0:def 4;
      hence contradiction by A5,TARSKI:def 1;
    end;
    hence thesis by A6,MENELAUS:13;
  end;
