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 Th21:
  K is being_line & P is being_line & Q is being_line & Q c= Plane
  (K,P) implies P // Q or ex q st q in P & q in Q
proof
  assume that
A1: K is being_line and
A2: P is being_line and
A3: Q is being_line and
A4: Q c= Plane(K,P);
  consider a,b such that
A5: a in Q and
A6: b in Q and
A7: a<>b by A3,AFF_1:19;
  consider a9 such that
A8: a,a9 // K and
A9: a9 in P by A4,A5,Lm3;
  consider A such that
A10: a9 in A and
A11: K // A by A1,AFF_1:49;
A12: a9,a // A by A8,A11,Th3;
  then
A13: a in A by A10,Th2;
  consider b9 such that
A14: b,b9 // K and
A15: b9 in P by A4,A6,Lm3;
  consider M such that
A16: b9 in M and
A17: K // M by A1,AFF_1:49;
A18: b9,b // M by A14,A17,Th3;
  then
A19: b in M by A16,Th2;
A20: A is being_line by A11,AFF_1:36;
A21: now
    assume A=M;
    then
A22: b in A by A16,A18,Th2;
    a in A by A10,A12,Th2;
    then a9 in Q by A3,A5,A6,A7,A10,A20,A22,AFF_1:18;
    hence ex q st q in P & q in Q by A9;
  end;
  A // M by A11,A17,AFF_1:44;
  hence thesis by A2,A3,A5,A6,A9,A15,A10,A16,A13,A19,A21,Th18;
end;
