reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem
  p in M & a in M & b in M & p in N & a9 in N & b9 in N & not p in P &
  not p in Q & M<>N & a in P & a9 in P & b in Q & b9 in Q & M is being_line & N
is being_line & P is being_line & Q is being_line implies (P // Q or ex q st q
  in P & q in Q)
proof
  assume that
A1: p in M and
A2: a in M and
A3: b in M and
A4: p in N and
A5: a9 in N and
A6: b9 in N and
A7: not p in P and
A8: not p in Q and
A9: M<>N and
A10: a in P and
A11: a9 in P and
A12: b in Q and
A13: b9 in Q and
A14: M is being_line and
A15: N is being_line and
A16: P is being_line and
A17: Q is being_line;
A18: a<>a9 by A1,A2,A4,A5,A7,A9,A10,A14,A15,AFF_1:18;
  LIN p,a,b by A1,A2,A3,A14,AFF_1:21;
  then consider c such that
A19: LIN p,a9,c and
A20: a,a9 // b,c by A7,A10,Th1;
  set D=Line(b,c);
A21: b in D by AFF_1:15;
A22: c in D by AFF_1:15;
A23: b<>b9 by A1,A3,A4,A6,A8,A9,A12,A14,A15,AFF_1:18;
A24: c in N by A4,A5,A7,A11,A15,A19,AFF_1:25;
  then
A25: b<>c by A1,A3,A4,A8,A9,A12,A14,A15,AFF_1:18;
  then
A26: D is being_line by AFF_1:def 3;
  now
    assume D<>Q;
    then
A27: c <>b9 by A12,A13,A17,A23,A26,A21,A22,AFF_1:18;
    LIN b9,c,a9 by A5,A6,A15,A24,AFF_1:21;
    then consider q such that
A28: LIN b9,b,q and
A29: c,b // a9,q by A27,Th1;
    a9,a // c,b by A20,AFF_1:4;
    then a9,a // a9,q by A25,A29,AFF_1:5;
    then LIN a9,a,q by AFF_1:def 1;
    then
A30: q in P by A10,A11,A16,A18,AFF_1:25;
    q in Q by A12,A13,A17,A23,A28,AFF_1:25;
    hence ex q st q in P & q in Q by A30;
  end;
  hence thesis by A10,A11,A12,A16,A17,A18,A20,A25,A22,AFF_1:38;
end;
