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

theorem Th15:
  A,B,C is_a_triangle & angle (C,B,G) = angle (C,B,A)/3 &
  angle (G,C,B) = angle (A,C,B)/3 implies C,G,B is_a_triangle
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: angle (C,B,G) = angle (C,B,A)/3 and
A3: angle (G,C,B) = angle (A,C,B)/3;
A4: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
A5: angle(A,C,B) <> 0 & angle(C,B,A) <> 0 by A1,EUCLID10:30;
    now
      thus C,G,B are_mutually_distinct by A2,A3,A4,A5,Th1;
      hereby
        assume angle(C,G,B)=PI;
        then G in LSeg(C,B) & G <> C by A2,A5,Th1,EUCLID_6:11;
        then angle(G,C,B) = angle(B,C,B) by EUCLID_6:9
                         .= 0 by Th1;
        hence contradiction by A3,A1,EUCLID10:30;
      end;
      hereby
        assume angle(G,B,C)=PI;
        then
A6:     angle(C,B,G) = 2*PI - PI by COMPTRIG:5,EUCLID_3:37
                    .= PI;
        2*PI+0 < 2*PI+PI by COMPTRIG:5,XREAL_1:8;
        hence contradiction by A2,A6,Th2;
      end;
      hereby
        assume angle(B,C,G)=PI;
        then
A7:     angle(G,C,B) = 2*PI - PI by COMPTRIG:5,EUCLID_3:37
                    .= PI;
        2*PI+0 < 2*PI+PI by COMPTRIG:5,XREAL_1:8;
        hence contradiction by A3,A7,Th2;
      end;
    end;
    hence thesis by EUCLID_6:20;
  end;
