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) = PI implies angle(B,C,A) = 0 &
  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) = PI;
    set z1 = euc2cpx(A);
    set z2 = euc2cpx(B);
    set z3 = euc2cpx(C);
    z1 <> z2 & z2 <> z3 & z1 <> z3 & angle(z1,z2,z3) = PI
    by A1,A2,EUCLID_3:4,EUCLID_3:def 4;
    then angle(z2,z3,z1) = 0 & angle(z3,z1,z2) = 0 by COMPLEX2:86;
    hence angle(B,C,A) = 0 & angle(C,A,B) = 0 by EUCLID_3:def 4;
    hence thesis by A2;
  end;
