reserve A,B,C,D,E,F,G for Point of TOP-REAL 2;

theorem Th14:
  A,B,C is_a_triangle & angle (A,C,F) = angle (A,C,B) / 3 &
  angle (F,A,C) = angle (B,A,C) / 3 implies A,F,C is_a_triangle
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: angle (A,C,F) = angle (A,C,B) / 3 and
A3: angle (F,A,C) = angle (B,A,C) / 3;
A4: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
A5: angle(A,C,B) <> 0 & angle(B,A,C) <> 0 by A1,EUCLID10:30;
    now
      thus A,F,C are_mutually_distinct by A2,A3,A4,A5,Th1;
      hereby
        assume angle(A,F,C)=PI;
        then F in LSeg(A,C) & F <> A by A2,A5,Th1,EUCLID_6:11;
        then angle(C,A,F) = angle(F,A,F) by EUCLID_6:9
        .= 0 by Th1;
        hence contradiction by A5,A3,EUCLID_3:36;
      end;
      hereby
        assume
A6:     angle(C,A,F)=PI;
        then
A7:     angle(F,A,C) = 2*PI - angle(C,A,F) by COMPTRIG:5,EUCLID_3:37
                    .= PI by A6;
        2*PI+0 < 2*PI+PI by XREAL_1:8,COMPTRIG:5;
        hence contradiction by A3,A7,Th2;
      end;
      hereby
        assume
A8:     angle(F,C,A)=PI;
        then
A9:     angle(A,C,F) = 2*PI - angle(F,C,A) by COMPTRIG:5,EUCLID_3:37
                    .= PI by A8;
        2*PI+0<2*PI+PI by COMPTRIG:5,XREAL_1:8;
        hence contradiction by A9,A2,Th2;
      end;
    end;
    hence thesis by EUCLID_6:20;
  end;
