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

theorem Th13:
  A,B,C is_a_triangle & angle (E,B,A) = angle (C,B,A) / 3 &
  angle (B,A,E) = angle (B,A,C) / 3 implies A,B,E is_a_triangle
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: angle (E,B,A) = angle (C,B,A) / 3 and
A3: angle (B,A,E) = angle (B,A,C) / 3;
A4: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
A5: angle(C,B,A) <> 0 & angle(B,A,C) <> 0 by A1,EUCLID10:30;
    now
      thus A,B,E are_mutually_distinct by A4,A2,Th1,A3,A5;
      hereby
        assume angle(A,B,E)=PI;
        then B in LSeg(A,E) & B <> A by A4,EUCLID_6:11;
        then angle(E,A,B) = angle(B,A,B) by EUCLID_6:9
                         .= 0 by Th1;
        hence contradiction by A5,A3,EUCLID_3:36;
      end;
      hereby assume angle(B,E,A)=PI;
        then E in LSeg(B,A) & E <> B by Th1,A3,A5,EUCLID_6:11;
        then angle(A,B,E) = angle(E,B,E) by EUCLID_6:9
                         .= 0 by Th1;
        hence contradiction by A5,A2,EUCLID_3:36;
      end;
      hereby
        assume
A6:     angle(E,A,B)=PI;
        then
A7:     angle(B,A,E) = 2*PI - angle(E,A,B) by COMPTRIG:5,EUCLID_3:37
                    .= PI by A6;
        2*PI+0 < 2*PI+PI by XREAL_1:8,COMPTRIG:5;
        hence contradiction by A7,A3,Th2;
      end;
    end;
    hence thesis by EUCLID_6:20;
  end;
