let FT be non empty RelStr ; :: thesis: for A being Subset of FT holds A ^delta = (A ^b ) \ (A ^i )
let A be Subset of FT; :: thesis: A ^delta = (A ^b ) \ (A ^i )
for x being set holds
( x in A ^delta iff x in (A ^b ) \ (A ^i ) )

proof

let x be set ; :: thesis: ( x in A ^delta iff x in (A ^b ) \ (A ^i ) )
thus ( x in A ^delta implies x in (A ^b ) \ (A ^i ) ) :: thesis: ( x in (A ^b ) \ (A ^i ) implies x in A ^delta )

proof

assume x in A ^delta ; :: thesis: x in (A ^b ) \ (A ^i )
then x in (A ^b ) /\ ((A ^i ) ` ) by Th2;
hence x in (A ^b ) \ (A ^i ) by SUBSET_1:32; :: thesis: verum

end;

assume x in (A ^b ) \ (A ^i ) ; :: thesis: x in A ^delta
then x in (A ^b ) /\ ((A ^i ) ` ) by SUBSET_1:32;
hence x in A ^delta by Th2; :: thesis: verum

end;

hence A ^delta = (A ^b ) \ (A ^i ) by TARSKI:2; :: thesis: verum