theorem Th12:
  A is being_line implies ex q st not q in A
proof
  assume
A1: A is being_line;
  then consider a,b such that
A2: a in A & b in A and
A3: a<>b by AFF_1:19;
  consider q such that
A4: not LIN a,b,q by A3,AFF_1:13;
  not q in A by A1,A2,A4,AFF_1:21;
  hence thesis;
end;
