let FT be non empty RelStr ; :: thesis: for A being Subset of FT st FT is reflexive & FT is symmetric & FT is connected & A <> {} & A ` <> {} holds
A ^deltao <> {}
let A be Subset of FT; :: thesis: ( FT is reflexive & FT is symmetric & FT is connected & A <> {} & A ` <> {} implies A ^deltao <> {} )
assume A1: ( FT is reflexive & FT is symmetric & FT is connected & A <> {} & A ` <> {} ) ; :: thesis: A ^deltao <> {}
A2: {} = {} FT ;
A \/ (A ` ) = [#] FT by XBOOLE_1:45;
then A3: not A,A ` are_separated by A1, A2, Th5, XBOOLE_1:79;

A4: now

assume A ^b meets A ` ; :: thesis: A ^deltao <> {}
then consider x being set such that
A5: ( x in A ^b & x in A ` ) by XBOOLE_0:3;
reconsider x = x as Element of FT by A5;
A6: U_FT x meets A by A5, FIN_TOPO:13;
x in U_FT x by A1, FIN_TOPO:def 4;
then U_FT x meets A ` by A5, XBOOLE_0:3;
then x in A ^delta by A6;
hence A ^deltao <> {} by A5, XBOOLE_0:def 4; :: thesis: verum

end;

now

assume A meets (A ` ) ^b ; :: thesis: A ^deltao <> {}
then consider x being set such that
A7: ( x in (A ` ) ^b & x in A ) by XBOOLE_0:3;
reconsider x = x as Element of FT by A7;
U_FT x meets A ` by A7, FIN_TOPO:13;
then consider y being set such that
A8: ( y in U_FT x & y in A ` ) by XBOOLE_0:3;
reconsider y = y as Element of FT by A8;
A9: x in U_FT y by A1, A8, FIN_TOPO:def 15;
y in U_FT y by A1, FIN_TOPO:def 4;
then A10: U_FT y meets A ` by A8, XBOOLE_0:3;
U_FT y meets A by A7, A9, XBOOLE_0:3;
then y in A ^delta by A10;
hence A ^deltao <> {} by A8, XBOOLE_0:def 4; :: thesis: verum

end;

hence A ^deltao <> {} by A3, A4, FINTOPO4:def 1; :: thesis: verum