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 Th18:
  a in M & b in M & a9 in N & b9 in N & a in P & a9 in P & b in Q
  & b9 in Q & M<>N & M // N & 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: a in M and
A2: b in M and
A3: a9 in N and
A4: b9 in N and
A5: a in P and
A6: a9 in P and
A7: b in Q and
A8: b9 in Q and
A9: M<>N and
A10: M // N and
A11: P is being_line and
A12: Q is being_line;
A13: a<>a9 by A1,A3,A9,A10,AFF_1:45;
A14: N is being_line by A10,AFF_1:36;
A15: b<>b9 by A2,A4,A9,A10,AFF_1:45;
A16: M is being_line by A10,AFF_1:36;
  now
    assume
A17: a<>b;
    consider c such that
A18: a,b // a9,c and
A19: a,a9 // b,c by DIRAF:40;
    set D=Line(b,c);
A20: b in D by AFF_1:15;
A21: c in D by AFF_1:15;
    a,b // N by A1,A2,A10,A16,AFF_1:43,52;
    then a9,c // N by A17,A18,AFF_1:32;
    then
A22: c in N by A3,A14,AFF_1:23;
    then
A23: b<>c by A2,A9,A10,AFF_1:45;
    then
A24: D is being_line by AFF_1:def 3;
    now
      assume D<>Q;
      then
A25:  c <>b9 by A7,A8,A12,A15,A24,A20,A21,AFF_1:18;
      LIN b9,c,a9 by A3,A4,A14,A22,AFF_1:21;
      then consider q such that
A26:  LIN b9,b,q and
A27:  c,b // a9,q by A25,Th1;
      a9,a // c,b by A19,AFF_1:4;
      then a9,a // a9,q by A23,A27,AFF_1:5;
      then LIN a9,a,q by AFF_1:def 1;
      then
A28:  q in P by A5,A6,A11,A13,AFF_1:25;
      q in Q by A7,A8,A12,A15,A26,AFF_1:25;
      hence ex q st q in P & q in Q by A28;
    end;
    hence thesis by A5,A6,A7,A11,A12,A13,A19,A23,A21,AFF_1:38;
  end;
  hence thesis by A5,A7;
end;
