let FT be non empty RelStr ; :: thesis: for A being Subset of FT st FT is filled & FT is symmetric & FT is connected & A <> {} & A ` <> {} holds
A ^deltao <> {}
let A be Subset of FT; :: thesis: ( FT is filled & FT is symmetric & FT is connected & A <> {} & A ` <> {} implies A ^deltao <> {} )
assume that
A1: ( FT is filled & FT is symmetric ) and
A2: ( FT is connected & A <> {} & A ` <> {} ) ; :: thesis: A ^deltao <> {}

A3: now :: thesis: ( A meets (A `) ^b implies A ^deltao <> {} )

assume A meets (A `) ^b ; :: thesis: A ^deltao <> {}
then consider x being object such that
A4: x in (A `) ^b and
A5: x in A by XBOOLE_0:3;
reconsider x = x as Element of FT by A4;
U_FT x meets A ` by A4, FIN_TOPO:8;
then consider y being object such that
A6: y in U_FT x and
A7: y in A ` by XBOOLE_0:3;
reconsider y = y as Element of FT by A6;
y in U_FT y by A1;
then A8: U_FT y meets A ` by A7, XBOOLE_0:3;
x in U_FT y by A1, A6;
then U_FT y meets A by A5, XBOOLE_0:3;
then y in A ^delta by A8;
hence A ^deltao <> {} by A7, XBOOLE_0:def 4; :: thesis: verum

end;

A9: now :: thesis: ( A ^b meets A ` implies A ^deltao <> {} )

assume A ^b meets A ` ; :: thesis: A ^deltao <> {}
then consider x being object such that
A10: x in A ^b and
A11: x in A ` by XBOOLE_0:3;
reconsider x = x as Element of FT by A10;
x in U_FT x by A1;
then A12: U_FT x meets A ` by A11, XBOOLE_0:3;
U_FT x meets A by A10, FIN_TOPO:8;
then x in A ^delta by A12;
hence A ^deltao <> {} by A11, XBOOLE_0:def 4; :: thesis: verum

end;

( {} = {} FT & A \/ (A `) = [#] FT ) by XBOOLE_1:45;
then not A,A ` are_separated by A2, Th4, XBOOLE_1:79;
hence A ^deltao <> {} by A9, A3, FINTOPO4:def 1; :: thesis: verum