let AS be AffinSpace; :: thesis: for a being Element of AS
for Q, K, P being Subset of AS st a in Q & a in Plane (K,P) & K // Q holds
Q c= Plane (K,P)
let a be Element of AS; :: thesis: for Q, K, P being Subset of AS st a in Q & a in Plane (K,P) & K // Q holds
Q c= Plane (K,P)
let Q, K, P be Subset of AS; :: thesis: ( a in Q & a in Plane (K,P) & K // Q implies Q c= Plane (K,P) )
assume that
A1: a in Q and
A2: a in Plane (K,P) and
A3: K // Q ; :: thesis: Q c= Plane (K,P)

now

A4: Plane (K,P) = Plane (Q,P) by A3, Th16;
let x be set ; :: thesis: ( x in Q implies x in Plane (K,P) )
assume A5: x in Q ; :: thesis: x in Plane (K,P)
reconsider c = x as Element of AS by A5;
A6: Q is being_line by A3, AFF_1:36;
consider b being Element of AS such that
A7: a,b // K and
A8: b in P by A2, Lm3;
a,b // Q by A3, A7, AFF_1:43;
then b in Q by A1, A6, AFF_1:23;
then c,b // Q by A5, A6, AFF_1:23;
hence x in Plane (K,P) by A8, A4; :: thesis: verum

end;

hence Q c= Plane (K,P) by TARSKI:def 3; :: thesis: verum