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 Th48:
  AS is not AffinPlane & X is being_plane implies ex q st not q in X
proof
  assume that
A1: AS is not AffinPlane and
A2: X is being_plane;
  assume
A3: not ex q st not q in X;
  for a,b,c,d st not a,b // c,d ex q st a,b // a,q & c,d // c,q
  proof
    let a,b,c,d;
    set M=Line(a,b),N=Line(c,d);
A4: a in M & b in M by AFF_1:15;
A5: c in N & d in N by AFF_1:15;
    assume
A6: not a,b // c,d;
    then
A7: a<>b by AFF_1:3;
    then
A8: M is being_line by AFF_1:def 3;
A9: c <>d by A6,AFF_1:3;
    then
A10: N is being_line by AFF_1:def 3;
    c in X & d in X by A3;
    then
A11: N c= X by A2,A9,Th19;
    a in X & b in X by A3;
    then M c= X by A2,A7,Th19;
    then consider q such that
A12: q in M and
A13: q in N by A2,A6,A11,A8,A10,A4,A5,Th22,AFF_1:39;
    LIN c,d,q by A10,A5,A13,AFF_1:21;
    then
A14: c,d // c, q by AFF_1:def 1;
    LIN a,b,q by A8,A4,A12,AFF_1:21;
    then a,b // a,q by AFF_1:def 1;
    hence thesis by A14;
  end;
  hence contradiction by A1,DIRAF:def 7;
end;
