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

theorem Th10:
  C,A,B is_a_triangle & angle(A,C,B) < PI &
  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
  (PI/3 + angle(A,C,B)/3) + (PI/3 + angle(C,B,A)/3) + angle(E,A,F) = PI
  proof
    assume that
A1: C,A,B is_a_triangle and
A2: angle(A,C,B)<PI and
A3: A,F,C is_a_triangle and
A4: F,A,E is_a_triangle and
A5: E,A,B is_a_triangle and
A6: angle (B,A,E) = angle (B,A,C) / 3 and
A7: angle (F,A,C) = angle (B,A,C)/3;
    set lambda = (PI/3 + angle(A,C,B)/3) + (PI/3 + angle(C,B,A)/3)
                   + angle(E,A,F);
    set theta  = angle(A,C,B)/3 + angle(C,B,A)/3 + angle(B,A,C)/3;
A8: angle(E,A,F) = angle(B,A,C)/3 by A1,A3,A4,A5,A6,A7,Th9;
    C,A,B are_mutually_distinct & angle(A,C,B) < PI by A1,A2,EUCLID_6:20;
    then angle(A,C,B) + angle(C,B,A) + angle(B,A,C) = PI by EUCLID_3:47;
    hence thesis by A8;
end;
