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);

then [a,b] in EqRel (R,I) by Def5;

hence a in Class ((EqRel (R,I)),b) by EQREL_1:19; :: thesis: verum

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 - b in I
; :: thesis: 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_1:19;

hence a - b in I by Def5; :: thesis: verum

end;then [a,b] in EqRel (R,I) by EQREL_1:19;

hence a - b in I by Def5; :: thesis: verum

then [a,b] in EqRel (R,I) by Def5;

hence a in Class ((EqRel (R,I)),b) by EQREL_1:19; :: thesis: verum