let G be TopologicalGroup; :: 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 Th41;
reconsider B = X /\ Y as open a_neighborhood of a by CONNSP_2:4, TOPS_1:38;
take B ; :: thesis: B * (B " ) c= A
let x be set ; :: 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 & g in B & h in B " ) ;
h " in B by A2, Th7;
then h " in Y by XBOOLE_0:def 4;
then ( g in X & h in Y " ) by A2, Th7, XBOOLE_0:def 4;
then x in X * (Y " ) by A2;
hence x in A by A1; :: thesis: verum