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

theorem Th11:
  A,C,B is_a_triangle implies sin (PI/3 - angle(A,C,B)/3)<>0
  proof
    assume
A1: A,C,B is_a_triangle;
    assume sin (PI/3 - angle(A,C,B)/3)=0;
    then consider i0 be Integer such that
A2: PI/3 - angle(A,C,B)/3 = PI * i0 by BORSUK_7:7;
    0 + 3 * i0 * PI <= PI - 3 * i0 * PI + (3 * i0 * PI) &
    PI - 3 * i0 *PI + (3 * i0 * PI) < 2 * PI + (3 * i0 * PI)
    by A2,XREAL_1:6,Th2;
    then 3 * i0 * PI <= PI & PI - PI < 2 * PI + 3 * i0 * PI - PI by XREAL_1:9;
    then 3 * i0 * PI / PI <= PI / PI & 0 < (1 + 3 * i0) * PI
    by COMPTRIG:5,XREAL_1:72;
    then 3 * i0 * (PI / PI) <= PI / PI & 0 / PI < (1 + 3 * i0) * PI / PI
    by COMPTRIG:5;
    then 3 * i0 <= PI / PI & 0 / PI < (1 + 3 * i0) by XCMPLX_1:88,COMPTRIG:5;
    then 3 * i0 <= 1 & 0 < (1 + 3 * i0) by XCMPLX_1:60;
    then i0 = 0 by Lm4;
    hence contradiction by A1,A2,EUCLID_6:20;
  end;
