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
  K // P implies Plane(K,P) = P
proof
  set X=Plane(K,P);
  assume
A1: K // P;
  then
A2: P is being_line by AFF_1:36;
  now
    let x be object;
    assume x in X;
    then consider a such that
A3: x=a and
A4: ex b st a,b // K & b in P;
    consider b such that
A5: a,b // K and
A6: b in P by A4;
    a,b // P by A1,A5,AFF_1:43;
    then b,a // P by AFF_1:34;
    hence x in P by A2,A3,A6,AFF_1:23;
  end;
  then
A7: X c= P;
  K is being_line by A1,AFF_1:36;
  then P c= X by Th14;
  hence thesis by A7,XBOOLE_0:def 10;
end;
