reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th85:
  a <> b & a <> c & b <> c & 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
proof
  assume that
A1: a <> b and
A2: a <> c and
A3: b <> c and
A4: angle(a,b,c) = 0;
  per cases by A1,A2,A3,A4,Th81;
  suppose
A5: angle(b,c,a) <> 0;
A6: 0 <= angle(b,c,a) by Th68;
    thus thesis
    proof
      per cases by XXREAL_0:1;
      suppose
        angle(b,c,a) < PI;
        hence thesis by A2,A3,A4,A5,A6,Th82;
      end;
      suppose
        angle(b,c,a) = PI;
        hence thesis by A2,A3,Th84;
      end;
      suppose
        angle(b,c,a) > PI;
        hence thesis by A2,A3,A4,Th83,COMPTRIG:5;
      end;
    end;
  end;
  suppose
A7: angle(c,a,b) <> 0;
A8: 0 <= angle(c,a,b) by Th68;
    thus thesis
    proof
      per cases by XXREAL_0:1;
      suppose
        angle(c,a,b) < PI;
        hence thesis by A1,A2,A4,A7,A8,Th82;
      end;
      suppose
        angle(c,a,b) = PI;
        hence thesis by A1,A2,Th84;
      end;
      suppose
        angle(c,a,b) > PI;
        hence thesis by A1,A2,A4,Th83,COMPTRIG:5;
      end;
    end;
  end;
end;
