let AS be AffinSpace; :: thesis: for a being Element of AS
for A being Subset of AS st A is being_line holds
( a in A iff a * A = A )
let a be Element of AS; :: thesis: for A being Subset of AS st A is being_line holds
( a in A iff a * A = A )
let A be Subset of AS; :: thesis: ( A is being_line implies ( a in A iff a * A = A ) )
assume A1: A is being_line ; :: thesis: ( a in A iff a * A = A )

now :: thesis: ( a in A implies a * A = A )

assume A2: a in A ; :: thesis: a * A = A
A // A by A1, AFF_1:41;
hence a * A = A by A1, A2, Def3; :: thesis: verum

end;

hence ( a in A iff a * A = A ) by A1, Def3; :: thesis: verum