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

for P being PLANE of S holds

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

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

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

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

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

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

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

hence C on P by A2, TARSKI:def 2; :: thesis: verum

for P being PLANE of S holds

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

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

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

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

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

proof

assume A2:
( A on P & B on P )
; :: thesis: {A,B} on P
A1:
( A in {A,B} & B in {A,B} )
by TARSKI:def 2;

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

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

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

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

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

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

hence C on P by A2, TARSKI:def 2; :: thesis: verum