reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;

theorem Th16:
  the carrier of AS misses Dir_of_Lines(AS)
proof
  assume not thesis;
  then consider x being object such that
A1: x in (the carrier of AS) and
A2: x in Dir_of_Lines(AS) by XBOOLE_0:3;
  reconsider a=x as Element of AS by A1;
  consider X such that
A3: x=LDir(X) and
A4: X is being_line by A2,Th14;
  set A=a*X;
A5: A is being_line by A4,AFF_4:27;
  X // A by A4,AFF_4:def 3;
  then X '||' A by A4,A5,AFF_4:40;
  then A in a by A3,A4,A5,Th9;
  hence contradiction by A4,AFF_4:def 3;
end;
