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

theorem Th7:
  A,F,C is_a_triangle & angle (C,F,A) < PI &
  angle (A,C,F) = angle (A,C,B)/3 & angle (F,A,C) = angle (B,A,C)/3 &
  angle (A,C,B)/3 + angle(B,A,C)/3 + angle(C,B,A)/3 = PI/3
  implies
  |.A-F.| * sin (PI/3 - angle(C,B,A)/3) = |.A-C.| * sin (angle (A,C,B)/3)
  proof
    assume that
A1: A,F,C is_a_triangle and
A2: angle (C,F,A) < PI and
A3: angle (A,C,F) = angle (A,C,B)/3 and
A4: angle (F,A,C) = angle (B,A,C)/3 and
A5: angle (A,C,B)/3 + angle(B,A,C)/3 + angle(C,B,A)/3 = PI/3;
A6: angle (F,C,A) = 2*PI - angle(A,C,B)/3 by A1,A3,EUCLID10:31;
A7: angle(A,C,F) = 2*PI - angle(F,C,A) &
    angle(F,A,C) = 2*PI - angle(C,A,F) by A1,EUCLID10:31;
A8: A,F,C are_mutually_distinct by A1,EUCLID_6:20;
    F,A,C is_a_triangle & angle(C,F,A)<PI by A1,A2,MENELAUS:15;
    then angle(C,A,F) + angle(A,F,C) + angle(F,C,A) = 5 * PI by EUCLID10:54;
    then angle(A,F,C)=4*PI/3 - angle(C,B,A)/3 by A7,A3,A5,A4;
    then
A9: sin angle(A,F,C) = - sin (PI/3 - angle(C,B,A)/3) by Lm5;
    |.A-F.| * sin angle(A,F,C) = |.A-C.| * sin (2*PI - angle (A,C,B)/3)
    by A8,A6,EUCLID_6:6;
    then |.A-F.| * - sin (PI/3 - angle(C,B,A)/3)
    = |.A-C.| * (-sin (angle (A,C,B)/3)) by A9,EUCLID10:3;
    hence thesis;
  end;
