reserve SAS for AffinPlane;

theorem Th2:
  SAS is Desarguesian implies for o,a,a9,b,b9,c,c9 being Element of
  SAS holds ( not o,a // o,b & not o,a // o,c & o,a // o,a9 & o,b // o,b9 & o,c
  // o,c9 & a,b // a9,b9 & a,c // a9,c9 implies b,c // b9,c9 )
proof
  assume
A1: SAS is Desarguesian;
  let o,a,a9,b,b9,c,c9 be Element of SAS such that
A2: not o,a // o,b and
A3: not o,a // o,c and
A4: o,a // o,a9 and
A5: o,b // o,b9 and
A6: o,c // o,c9 and
A7: a,b // a9,b9 & a,c // a9,c9;
A8: o<>a & o<>b by A2,AFF_1:3;
A9: o<>c by A3,AFF_1:3;
  LIN o,b,b9 by A5,AFF_1:def 1;
  then consider P being Subset of SAS such that
A10: P is being_line & o in P and
A11: b in P and
A12: b9 in P by AFF_1:21;
  LIN o,a,a9 by A4,AFF_1:def 1;
  then consider A being Subset of SAS such that
A13: A is being_line & o in A & a in A and
A14: a9 in A by AFF_1:21;
  LIN o,c,c9 by A6,AFF_1:def 1;
  then consider C being Subset of SAS such that
A15: C is being_line & o in C and
A16: c in C and
A17: c9 in C by AFF_1:21;
A18: A<>C by A3,A13,A16,AFF_1:51;
  A<>P by A2,A13,A11,AFF_1:51;
  hence thesis by A1,A7,A13,A14,A10,A11,A12,A15,A16,A17,A8,A9,A18,AFF_2:def 4;
end;
