let S be IncStruct ; :: thesis: for A, B, C, D being POINT of S
for P being PLANE of S holds
( {A,B,C,D} on P iff ( A on P & B on P & C on P & D on P ) )
let A, B, C, D be POINT of S; :: thesis: for P being PLANE of S holds
( {A,B,C,D} on P iff ( A on P & B on P & C on P & D on P ) )
let P be PLANE of S; :: thesis: ( {A,B,C,D} on P iff ( A on P & B on P & C on P & D on P ) )
thus ( {A,B,C,D} on P implies ( A on P & B on P & C on P & D on P ) ) :: thesis: ( A on P & B on P & C on P & D on P implies {A,B,C,D} on P )

proof

A1: ( C in {A,B,C,D} & D in {A,B,C,D} ) by ENUMSET1:def 2;
A2: ( A in {A,B,C,D} & B in {A,B,C,D} ) by ENUMSET1:def 2;
assume {A,B,C,D} on P ; :: thesis: ( A on P & B on P & C on P & D on P )
hence ( A on P & B on P & C on P & D on P ) by A2, A1; :: thesis: verum

end;

assume A3: ( A on P & B on P & C on P & D on P ) ; :: thesis: {A,B,C,D} on P
let E be POINT of S; :: according to INCSP_1:def 5 :: thesis: ( E in {A,B,C,D} implies E on P )
assume E in {A,B,C,D} ; :: thesis: E on P
hence E on P by A3, ENUMSET1:def 2; :: thesis: verum