let FT be non empty RelStr ; :: thesis: for A being Subset of FT holds A ^delta = (A ^deltai) \/ (A ^deltao)
let A be Subset of FT; :: thesis: A ^delta = (A ^deltai) \/ (A ^deltao)
for x being object holds
( x in A ^delta iff x in (A ^deltai) \/ (A ^deltao) )

proof

let x be object ; :: thesis: ( x in A ^delta iff x in (A ^deltai) \/ (A ^deltao) )
thus ( x in A ^delta implies x in (A ^deltai) \/ (A ^deltao) ) :: thesis: ( x in (A ^deltai) \/ (A ^deltao) implies x in A ^delta )

proof

assume x in A ^delta ; :: thesis: x in (A ^deltai) \/ (A ^deltao)
then x in ([#] the carrier of FT) /\ (A ^delta) by XBOOLE_1:28;
then x in (A \/ (A `)) /\ (A ^delta) by SUBSET_1:10;
hence x in (A ^deltai) \/ (A ^deltao) by XBOOLE_1:23; :: thesis: verum

end;

assume x in (A ^deltai) \/ (A ^deltao) ; :: thesis: x in A ^delta
then x in (A \/ (A `)) /\ (A ^delta) by XBOOLE_1:23;
then x in ([#] the carrier of FT) /\ (A ^delta) by SUBSET_1:10;
hence x in A ^delta by XBOOLE_1:28; :: thesis: verum

end;

hence A ^delta = (A ^deltai) \/ (A ^deltao) by TARSKI:2; :: thesis: verum