let S be IncProjStr ; :: thesis: for L being LINE of S
for A, B, C being POINT of S holds
( {A,B,C} on L iff ( A on L & B on L & C on L ) )
let L be LINE of S; :: thesis: for A, B, C being POINT of S holds
( {A,B,C} on L iff ( A on L & B on L & C on L ) )
let A, B, C be POINT of S; :: thesis: ( {A,B,C} on L iff ( A on L & B on L & C on L ) )
thus ( {A,B,C} on L implies ( A on L & B on L & C on L ) ) :: thesis: ( A on L & B on L & C on L implies {A,B,C} on L )
proof
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 L ; :: thesis: ( A on L & B on L & C on L )
hence ( A on L & B on L & C on L ) by A2, A1; :: thesis: verum
end;
assume A3: ( A on L & B on L & C on L ) ; :: thesis: {A,B,C} on L
let D be POINT of S; :: according to INCSP_1:def 4 :: thesis: ( D in {A,B,C} implies D on L )
assume D in {A,B,C} ; :: thesis: D on L
hence D on L by A3, ENUMSET1:def 1; :: thesis: verum