
theorem Th18:
  for OAS being OAffinSpace holds Lambda(OAS) is Fanoian
proof
  let OAS be OAffinSpace;
  set AS = Lambda(OAS);
  for a,b,c,d being Element of AS st a,b // c,d & a,c // b,d & a,d // b,c
  holds a,b // a,c
  proof
    let a,b,c,d be Element of AS such that
A1: a,b // c,d and
A2: a,c // b,d and
A3: a,d // b,c;
    reconsider a1=a,b1=b,c1=c,d1=d as Element of OAS by Th1;
    set P = Line(a,d),Q = Line(b,c);
    assume
A4: not a,b // a,c;
    then
A5: a<>d by A1,AFF_1:4;
    then
A6: P is being_line by AFF_1:def 3;
A7: not a1,b1,c1 are_collinear
    proof
      assume not thesis;
      then a1,b1 '||' a1,c1 by DIRAF:def 5;
      hence contradiction by A4,DIRAF:38;
    end;
    a1,b1 '||' c1,d1 & a1,c1 '||' b1,d1 by A1,A2,DIRAF:38;
    then consider x1 being Element of OAS such that
A8: x1,a1,d1 are_collinear and
A9: x1,b1,c1 are_collinear by A7,PASCH:25;
    reconsider x=x1 as Element of AS by Th1;
A10: d in P by AFF_1:15;
    x1,a1 '||' x1,d1 by A8,DIRAF:def 5;
    then x,a // x,d by DIRAF:38;
    then LIN x,a,d by AFF_1:def 1;
    then LIN a,d,x by AFF_1:6;
    then
A11: x in P by AFF_1:def 2;
A12: a in P & b in Q by AFF_1:15;
    x1,b1 '||' x1,c1 by A9,DIRAF:def 5;
    then x,b // x,c by DIRAF:38;
    then LIN x,b,c by AFF_1:def 1;
    then LIN b,c,x by AFF_1:6;
    then
A13: x in Q by AFF_1:def 2;
A14: c in Q by AFF_1:15;
A15: not LIN a,b,c by A4,AFF_1:def 1;
    then
A16: b<>c by AFF_1:7;
    then Q is being_line by AFF_1:def 3;
    then P // Q by A3,A16,A5,A6,A10,A12,A14,AFF_1:38;
    then P = Q by A11,A13,AFF_1:45;
    hence contradiction by A15,A6,A12,A14,AFF_1:21;
  end;
  hence thesis;
end;
