let AS be AffinSpace; :: thesis:
let K, M, P be Subset of AS; :: thesis:
assume A1: K // M ; :: thesis:

assume x in Plane (M,P) ; :: thesis:
then consider a being Element of AS such that
A3: x = a and
A4: ex b being Element of AS st
( a,b // M & b in P ) ;
consider b being Element of AS such that
A5: a,b // M and
A6: b in P by A4;
a,b // K by A1, A5, AFF_1:43;
hence x in Plane (K,P) by A3, A6; :: thesis:

end;

assume x in Plane (K,P) ; :: thesis:
then consider a being Element of AS such that
A7: x = a and
A8: ex b being Element of AS st
( a,b // K & b in P ) ;
consider b being Element of AS such that
A9: a,b // K and
A10: b in P by A8;
a,b // M by A1, A9, AFF_1:43;
hence x in Plane (M,P) by A7, A10; :: thesis:

end;

hence ( x in Plane (K,P) iff x in Plane (M,P) ) by A2; :: thesis:

end;