let X be non empty TopSpace; :: thesis: for x being Point of X
for V being Subset of X holds
( V is a_neighborhood of {x} iff V is a_neighborhood of x )
let x be Point of X; :: thesis: for V being Subset of X holds
( V is a_neighborhood of {x} iff V is a_neighborhood of x )
let V be Subset of X; :: thesis: ( V is a_neighborhood of {x} iff V is a_neighborhood of x )
thus ( V is a_neighborhood of {x} implies V is a_neighborhood of x ) :: thesis: ( V is a_neighborhood of x implies V is a_neighborhood of {x} )

proof

assume V is a_neighborhood of {x} ; :: thesis: V is a_neighborhood of x
then A1: {x} c= Int V by Def2;
x in {x} by TARSKI:def 1;
hence V is a_neighborhood of x by A1, Def1; :: thesis: verum

end;

assume V is a_neighborhood of x ; :: thesis: V is a_neighborhood of {x}
then x in Int V by Def1;
then for p being object st p in {x} holds
p in Int V by TARSKI:def 1;
then {x} c= Int V ;
hence V is a_neighborhood of {x} by Def2; :: thesis: verum