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 Th20:
  K is being_line & P is being_line & Q is being_line & not K // Q
  & Q c= Plane(K,P) implies Plane(K,Q) = Plane(K,P)
proof
  assume that
A1: K is being_line and
A2: P is being_line and
A3: Q is being_line and
A4: not K // Q and
A5: Q c= Plane(K,P);
A6: Plane(K,Q) c= Plane(K,P) by A1,A5,Lm6;
  consider a,b such that
A7: a in Q and
A8: b in Q and
A9: a<>b by A3,AFF_1:19;
  consider b9 such that
A10: b,b9 // K and
A11: b9 in P by A5,A8,Lm3;
  b9,b // K by A10,AFF_1:34;
  then
A12: b9 in Plane(K,Q) by A8;
  consider a9 such that
A13: a,a9 // K and
A14: a9 in P by A5,A7,Lm3;
A15: a9<>b9
  proof
    consider A such that
A16: a9 in A and
A17: K // A by A1,AFF_1:49;
    a9,a // A by A13,A17,Th3;
    then
A18: a in A by A16,Th2;
    assume a9=b9;
    then a9,b // A by A10,A17,Th3;
    then
A19: b in A by A16,Th2;
    A is being_line by A17,AFF_1:36;
    hence contradiction by A3,A4,A7,A8,A9,A17,A19,A18,AFF_1:18;
  end;
  a9,a // K by A13,AFF_1:34;
  then a9 in Plane(K,Q) by A7;
  then Plane(K,P) c= Plane(K,Q) by A1,A2,A3,A14,A11,A15,A12,Lm5,Lm6;
  hence thesis by A6,XBOOLE_0:def 10;
end;
