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 Th22:
  X is being_plane & M is being_line & N is being_line & M c= X &
  N c= X implies M // N or ex q st q in M & q in N
proof
  assume that
A1: X is being_plane and
A2: M is being_line and
A3: N is being_line and
A4: M c= X & N c= X;
  consider K,P such that
A5: K is being_line and
A6: P is being_line and
  not K // P and
A7: X = Plane(K,P) by A1;
A8: now
    assume not K // N;
    then M c= Plane(K,N) by A3,A4,A5,A6,A7,Th20;
    then N // M or ex q st q in N & q in M by A2,A3,A5,Th21;
    hence thesis;
  end;
  now
    assume not K // M;
    then N c= Plane(K,M) by A2,A4,A5,A6,A7,Th20;
    hence thesis by A2,A3,A5,Th21;
  end;
  hence thesis by A8,AFF_1:44;
end;
