let S be IncProjStr ; :: thesis: for L being LINE of S
for A, B being POINT of S holds
( {A,B} on L iff ( A on L & B on L ) )
let L be LINE of S; :: thesis: for A, B being POINT of S holds
( {A,B} on L iff ( A on L & B on L ) )
let A, B be POINT of S; :: thesis: ( {A,B} on L iff ( A on L & B on L ) )
thus ( {A,B} on L implies ( A on L & B on L ) ) :: thesis: ( A on L & B on L implies {A,B} on L )

proof

A1: ( A in {A,B} & B in {A,B} ) by TARSKI:def 2;
assume {A,B} on L ; :: thesis: ( A on L & B on L )
hence ( A on L & B on L ) by A1; :: thesis: verum

end;

assume A2: ( A on L & B on L ) ; :: thesis: {A,B} on L
let C be POINT of S; :: according to INCSP_1:def 4 :: thesis: ( C in {A,B} implies C on L )
assume C in {A,B} ; :: thesis: C on L
hence C on L by A2, TARSKI:def 2; :: thesis: verum