 reserve A, B, C, A1, B1, C1, A2, B2, C2, C3 for Point of TOP-REAL 2,
  lambda, mu, nu, alpha, beta, gamma for Real,
  X, Y, Z for Subset of TOP-REAL 2;

theorem
  Th8: (angle(A, B, C) = 0 & A, B, C are_mutually_distinct) implies
  (angle(B, C, A) = PI or angle(B, A, C) = PI)
proof
  set z1 = (euc2cpx A);
  set z2 = (euc2cpx B);
  set z3 = (euc2cpx C);
  assume that A1: angle(A, B, C) = 0 and A2: A, B, C are_mutually_distinct;
  A <> B & A <> C & B <> C by A2,ZFMISC_1:def 5;
  then A3: z1 <> z2 & z1 <> z3 & z2 <> z3 by EUCLID_3:4;
  per cases;
  suppose angle (z2, z3, z1) = 0 & angle (z3, z1, z2) = PI;
    hence thesis by COMPLEX2:82;
  end;
  suppose not (angle (z2, z3, z1) = 0 & angle (z3, z1, z2) = PI);
    hence thesis by A3,A1,COMPLEX2:87;
  end;
end;
