let R be Ring; :: thesis: for I being Ideal of R
for a, b being Element of R holds
( a in Class (EqRel R,I),b iff a - b in I )
let I be Ideal of R; :: thesis: for a, b being Element of R holds
( a in Class (EqRel R,I),b iff a - b in I )
let a, b be Element of R; :: thesis: ( a in Class (EqRel R,I),b iff a - b in I )
set E = EqRel R,I;

hereby :: thesis: ( a - b in I implies a in Class (EqRel R,I),b )

assume a in Class (EqRel R,I),b ; :: thesis: a - b in I
then [a,b] in EqRel R,I by EQREL_1:27;
hence a - b in I by Def5; :: thesis: verum

end;

assume a - b in I ; :: thesis: a in Class (EqRel R,I),b
then [a,b] in EqRel R,I by Def5;
hence a in Class (EqRel R,I),b by EQREL_1:27; :: thesis: verum