let G be TopologicaladdGroup; :: thesis: for a being Point of G
for A being a_neighborhood of a + (- a) ex B being open a_neighborhood of a st B + (- B) c= A
let a be Point of G; :: thesis: for A being a_neighborhood of a + (- a) ex B being open a_neighborhood of a st B + (- B) c= A
let A be a_neighborhood of a + (- a); :: thesis: ex B being open a_neighborhood of a st B + (- B) c= A
consider X, Y being open a_neighborhood of a such that
A1: X + (- Y) c= A by Th40;
reconsider B = X /\ Y as open a_neighborhood of a by CONNSP_2:2;
take B ; :: thesis: B + (- B) c= A
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in B + (- B) or x in A )
assume x in B + (- B) ; :: thesis: x in A
then consider g, h being Point of G such that
A2: x = g + h and
A3: g in B and
A4: h in - B ;
- h in B by A4, Th7;
then - h in Y by XBOOLE_0:def 4;
then A5: h in - Y by Th7;
g in X by A3, XBOOLE_0:def 4;
then x in X + (- Y) by A2, A5;
hence x in A by A1; :: thesis: verum