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

theorem Th9:
  C,A,B is_a_triangle & A,F,C is_a_triangle &
  F,A,E is_a_triangle & E,A,B is_a_triangle &
  angle(B,A,E) = angle(B,A,C) / 3 & angle(F,A,C) = angle(B,A,C) / 3
  implies angle(E,A,F) = angle(B,A,C) / 3
  proof
    assume that
A1: C,A,B is_a_triangle and
A2: A,F,C is_a_triangle and
A3: F,A,E is_a_triangle and
A4: E,A,B is_a_triangle and
A5: angle (B,A,E) = angle (B,A,C) / 3 and
A6: angle (F,A,C) = angle (B,A,C) / 3;
A7: angle(C,A,B)=2*PI-angle(B,A,C) by A1,EUCLID10:31;
A8: angle(C,A,F) + angle(F,A,E) = angle(C,A,E) or
    angle(C,A,F) + angle(F,A,E) = angle(C,A,E)+2*PI by EUCLID_6:4;
A9: angle(C,A,E)+angle(E,A,B)=angle(C,A,B) or
    angle(C,A,E)+angle(E,A,B)=angle(C,A,B)+2*PI by EUCLID_6:4;
A10: angle(C,A,F) = 2*PI-angle(B,A,C)/3 by A2,A6,EUCLID10:31;
A11: angle(F,A,E)=2*PI-angle(E,A,F) by A3,EUCLID10:31;
A12: angle(E,A,B) = 2*PI - angle(B,A,C)/3 by A5,A4,EUCLID10:31;
A13: not angle(E,A,F) = 4*PI + angle(B,A,C)/3
    proof
      assume
A14:  angle(E,A,F) = 4*PI + angle(B,A,C)/3;
      now
        0+ 2*PI < 2*PI + 2*PI by COMPTRIG:5,XREAL_1:8;
        hence 2*PI < 4*PI;
        0 <= angle(B,A,C) & angle(B,A,C) <> 0 by Th2,A1,EUCLID10:30;
        hence 0 < angle(B,A,C)/3;
      end;
      then 2*PI + 0 < 4*PI + angle(B,A,C)/3 by XREAL_1:8;
      hence contradiction by A14,Th2;
    end;
    not angle(E,A,F) = 2*PI + angle(B,A,C)/3
    proof
      assume
A15:  angle(E,A,F) = 2*PI + angle(B,A,C)/3;
      0 <= angle(B,A,C) & angle(B,A,C) <> 0 by Th2,A1,EUCLID10:30;
      then 2*PI + 0 < 2*PI + angle(B,A,C)/3 by XREAL_1:8;
      hence contradiction by A15,Th2;
    end;
    hence thesis by A12,A7,A8,A9,A10,A11,A13;
  end;
