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
  AS is AffinPlane implies AfLines(AS) misses Dir_of_Planes(AS)
proof
  the carrier of AS c= the carrier of AS;
  then reconsider X9=the carrier of AS as Subset of AS;
  assume AS is AffinPlane;
  then
A1: X9 is being_plane by Th1;
  assume not thesis;
  then consider x being object such that
A2: x in AfLines(AS) and
A3: x in Dir_of_Planes(AS) by XBOOLE_0:3;
  consider Y such that
A4: x=Y and
A5: Y is being_line by A2;
  consider X such that
A6: x=PDir(X) and
A7: X is being_plane by A3,Th15;
  consider a,b being Element of AS such that
A8: a in Y and
  b in Y and
  a<>b by A5,AFF_1:19;
  consider Y9 such that
A9: a = Y9 and
A10: Y9 is being_plane and
  X '||' Y9 by A6,A7,A4,A8,Th10;
A11: not Y9 in Y9;
  Y9 = X9 by A1,A10,AFF_4:33;
  hence contradiction by A9,A11;
end;
