consider F being Subset-Family of GX such that
A1: for A being Subset of GX holds
( A in F iff ( A is connected & x in A ) ) and
A2: Component_of x = union F by Def7;
A3: for A being set st A in F holds
x in A by A1;
F <> {} by A1, Th31;
then A4: meet F <> {} GX by A3, SETFAM_1:def 1;
for A being Subset of GX st A in F holds
A is connected by A1;
hence ( not Component_of x is empty & Component_of x is connected ) by A2, A4, Th26, Th38; :: thesis: verum