let AS be AffinSpace; :: thesis: for K, P being Subset of AS st K // P holds
Plane (K,P) = P
let K, P be Subset of AS; :: thesis: ( K // P implies Plane (K,P) = P )
set X = Plane (K,P);
assume A1: K // P ; :: thesis: Plane (K,P) = P
then A2: P is being_line by AFF_1:36;

now :: thesis: for x being object st x in Plane (K,P) holds
x in P

let x be object ; :: thesis: ( x in Plane (K,P) implies x in P )
assume x in Plane (K,P) ; :: thesis: x in P
then consider a being Element of AS such that
A3: x = a and
A4: ex b being Element of AS st
( a,b // K & b in P ) ;
consider b being Element of AS 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; :: thesis: verum

end;

then A7: Plane (K,P) c= P ;
K is being_line by A1, AFF_1:36;
then P c= Plane (K,P) by Th14;
hence Plane (K,P) = P by A7, XBOOLE_0:def 10; :: thesis: verum