let S be IncStruct ; :: thesis: for A, B, C being POINT of S

for P being PLANE of S holds

( {A,B,C} on P iff ( A on P & B on P & C on P ) )

let A, B, C be POINT of S; :: thesis: for P being PLANE of S holds

( {A,B,C} on P iff ( A on P & B on P & C on P ) )

let P be PLANE of S; :: thesis: ( {A,B,C} on P iff ( A on P & B on P & C on P ) )

thus ( {A,B,C} on P implies ( A on P & B on P & C on P ) ) :: thesis: ( A on P & B on P & C on P implies {A,B,C} on P )

let D be POINT of S; :: according to INCSP_1:def 5 :: thesis: ( D in {A,B,C} implies D on P )

assume D in {A,B,C} ; :: thesis: D on P

hence D on P by A3, ENUMSET1:def 1; :: thesis: verum

for P being PLANE of S holds

( {A,B,C} on P iff ( A on P & B on P & C on P ) )

let A, B, C be POINT of S; :: thesis: for P being PLANE of S holds

( {A,B,C} on P iff ( A on P & B on P & C on P ) )

let P be PLANE of S; :: thesis: ( {A,B,C} on P iff ( A on P & B on P & C on P ) )

thus ( {A,B,C} on P implies ( A on P & B on P & C on P ) ) :: thesis: ( A on P & B on P & C on P implies {A,B,C} on P )

proof

assume A3:
( A on P & B on P & C on P )
; :: thesis: {A,B,C} on P
A1:
C in {A,B,C}
by ENUMSET1:def 1;

A2: ( A in {A,B,C} & B in {A,B,C} ) by ENUMSET1:def 1;

assume {A,B,C} on P ; :: thesis: ( A on P & B on P & C on P )

hence ( A on P & B on P & C on P ) by A2, A1; :: thesis: verum

end;A2: ( A in {A,B,C} & B in {A,B,C} ) by ENUMSET1:def 1;

assume {A,B,C} on P ; :: thesis: ( A on P & B on P & C on P )

hence ( A on P & B on P & C on P ) by A2, A1; :: thesis: verum

let D be POINT of S; :: according to INCSP_1:def 5 :: thesis: ( D in {A,B,C} implies D on P )

assume D in {A,B,C} ; :: thesis: D on P

hence D on P by A3, ENUMSET1:def 1; :: thesis: verum