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

theorem
  A,B,C are_mutually_distinct & angle(A,B,C) = 0 implies (angle(B,C,A) =0 &
  angle(C,A,B) = PI) or (angle(B,C,A) = PI & angle(C,A,B) = 0) & angle(A,B,C)
  + angle(B,C,A) + angle(C,A,B) = PI
  proof
    assume that
A1: A,B,C are_mutually_distinct and
A2: angle(A,B,C) = 0;
    set z1 = euc2cpx(A);
    set z2 = euc2cpx(B);
    set z3 = euc2cpx(C);
    z1 <> z2 & z2 <> z3 & z1 <> z3 & angle(z1,z2,z3) = 0
    by A1,A2,EUCLID_3:4,EUCLID_3:def 4;
    then per cases by COMPLEX2:87;
    suppose angle(z2,z3,z1) = 0 & angle(z3,z1,z2) = PI;
      hence thesis by EUCLID_3:def 4;
    end;
    suppose angle(z2,z3,z1) = PI & angle(z3,z1,z2) = 0;
      then angle(B,C,A) = PI & angle(C,A,B) = 0 by EUCLID_3:def 4;
      hence thesis by A2;
    end;
  end;
