theorem Th12:
  X is being_line & Y is being_line implies (LDir(X)=LDir(Y) iff X '||' Y)
proof
  assume that
A1: X is being_line and
A2: Y is being_line;
  LDir(X)=LDir(Y) iff X // Y by A1,A2,Th11;
  hence thesis by A1,A2,AFF_4:40;
end;
