let A be non degenerated commutative Ring; for S being non empty Subset of A holds Ideals (A,S) = Ideals (A,(S -Ideal))
let S be non empty Subset of A; Ideals (A,S) = Ideals (A,(S -Ideal))
thus
Ideals (A,S) c= Ideals (A,(S -Ideal))
XBOOLE_0:def 10 Ideals (A,(S -Ideal)) c= Ideals (A,S)
let x be object ; TARSKI:def 3 ( not x in Ideals (A,(S -Ideal)) or x in Ideals (A,S) )
assume
x in Ideals (A,(S -Ideal))
; x in Ideals (A,S)
then consider y being Ideal of A such that
A8:
x = y
and
A9:
S -Ideal c= y
;
S c= S -Ideal
by IDEAL_1:def 14;
then
S c= y
by A9;
hence
x in Ideals (A,S)
by A8; verum