let R be non empty right_complementable left-distributive left_unital add-associative right_zeroed doubleLoopStr ; :: thesis: for I being Ideal of R
for a, b being Element of R holds
( Class ((EqRel (R,I)),a) = 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
( Class ((EqRel (R,I)),a) = Class ((EqRel (R,I)),b) iff a - b in I )
let a, b be Element of R; :: thesis: ( Class ((EqRel (R,I)),a) = Class ((EqRel (R,I)),b) iff a - b in I )
set E = EqRel (R,I);
thus ( Class ((EqRel (R,I)),a) = Class ((EqRel (R,I)),b) implies a - b in I ) :: thesis: ( a - b in I implies Class ((EqRel (R,I)),a) = Class ((EqRel (R,I)),b) )

proof

assume Class ((EqRel (R,I)),a) = Class ((EqRel (R,I)),b) ; :: thesis: a - b in I
then a in Class ((EqRel (R,I)),b) by EQREL_1:23;
hence a - b in I by Th5; :: thesis: verum

end;

assume a - b in I ; :: thesis: Class ((EqRel (R,I)),a) = Class ((EqRel (R,I)),b)
then a in Class ((EqRel (R,I)),b) by Th5;
hence Class ((EqRel (R,I)),a) = Class ((EqRel (R,I)),b) by EQREL_1:23; :: thesis: verum