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 object holds

( x in A ^delta iff x in (A ^b) \ (A ^i) )

let A be Subset of FT; :: thesis: A ^delta = (A ^b) \ (A ^i)

for x being object holds

( x in A ^delta iff x in (A ^b) \ (A ^i) )

proof

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

then x in (A ^b) /\ ((A ^i) `) by SUBSET_1:13;

hence x in A ^delta by Th2; :: thesis: verum

end;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 ^b) \ (A ^i)
; :: thesis: x in A ^delta
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_1:13; :: thesis: verum

end;then x in (A ^b) /\ ((A ^i) `) by Th2;

hence x in (A ^b) \ (A ^i) by SUBSET_1:13; :: thesis: verum

then x in (A ^b) /\ ((A ^i) `) by SUBSET_1:13;

hence x in A ^delta by Th2; :: thesis: verum