let AS be AffinSpace; :: thesis: for K, M, P being Subset of AS st K // M holds
Plane (K,P) = Plane (M,P)
let K, M, P be Subset of AS; :: thesis: ( K // M implies Plane (K,P) = Plane (M,P) )
assume A1: K // M ; :: thesis: Plane (K,P) = Plane (M,P)

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

let x be object ; :: thesis: ( x in Plane (K,P) iff x in Plane (M,P) )

A2: now :: thesis: ( x in Plane (M,P) implies x in Plane (K,P) )

assume x in Plane (M,P) ; :: thesis: x in Plane (K,P)
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: verum

end;

now :: thesis: ( x in Plane (K,P) implies x in Plane (M,P) )

assume x in Plane (K,P) ; :: thesis: x in Plane (M,P)
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: verum

end;

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

end;

hence Plane (K,P) = Plane (M,P) by TARSKI:2; :: thesis: verum