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

theorem Th81:
  a <> b & a <> c & b <> c implies angle(a,b,c) <> 0 or angle(b,c,
  a) <> 0 or angle(c,a,b) <> 0
proof
  assume that
A1: a <> b & a <> c and
A2: b <> c;
A3: b-a <> 0 & a-c <> 0 by A1;
  c-b <> 0 by A2;
  then
A4: Arg(c-b) < PI iff Arg (-(c-b)) >= PI by Th14;
A5: -(b-a) = a-b & -(a-c) = c-a;
A6: -(c-a) = a-c;
  assume
A7: not thesis;
  then
A8: Arg(a-c) = Arg (b-c) by Th72;
  Arg(b-a) = Arg (c-a) by A7,Th72;
  then Arg(a-b) = Arg (a-c) by A3,A5,A6,Th62;
  hence thesis by A7,A8,A4,Th72;
end;
