let X be BCI-algebra; :: thesis: for I being Ideal of X holds
( I is associative-ideal of X iff for x, y, z being Element of X st (x \ y) \ z in I holds
x \ (y \ z) in I )
let I be Ideal of X; :: thesis: ( I is associative-ideal of X iff for x, y, z being Element of X st (x \ y) \ z in I holds
x \ (y \ z) in I )
thus ( I is associative-ideal of X implies for x, y, z being Element of X st (x \ y) \ z in I holds
x \ (y \ z) in I ) :: thesis: ( ( for x, y, z being Element of X st (x \ y) \ z in I holds
x \ (y \ z) in I ) implies I is associative-ideal of X )

proof

assume A1: I is associative-ideal of X ; :: thesis: for x, y, z being Element of X st (x \ y) \ z in I holds
x \ (y \ z) in I
let x, y, z be Element of X; :: thesis: ( (x \ y) \ z in I implies x \ (y \ z) in I )
assume A2: (x \ y) \ z in I ; :: thesis: x \ (y \ z) in I
(x \ (x \ y)) \ y = 0. X by BCIALG_1:1;
then x \ (x \ y) <= y ;
then (x \ (x \ y)) \ (y \ z) <= y \ (y \ z) by BCIALG_1:5;
then A3: ((x \ (x \ y)) \ (y \ z)) \ z <= (y \ (y \ z)) \ z by BCIALG_1:5;
(y \ (y \ z)) \ z = 0. X by BCIALG_1:1;
then (((x \ (x \ y)) \ (y \ z)) \ z) \ (0. X) = 0. X by A3;
then ((x \ (x \ y)) \ (y \ z)) \ z = 0. X by BCIALG_1:2;
then ((x \ (x \ y)) \ (y \ z)) \ z in I by A1, Def4;
then ((x \ (y \ z)) \ (x \ y)) \ z in I by BCIALG_1:7;
hence x \ (y \ z) in I by A1, A2, Def4; :: thesis: verum

end;

assume A4: for x, y, z being Element of X st (x \ y) \ z in I holds
x \ (y \ z) in I ; :: thesis: I is associative-ideal of X
A5: for x, y, z being Element of X st (x \ y) \ z in I & y \ z in I holds
x in I

proof

let x, y, z be Element of X; :: thesis: ( (x \ y) \ z in I & y \ z in I implies x in I )
assume that
A6: (x \ y) \ z in I and
A7: y \ z in I ; :: thesis: x in I
x \ (y \ z) in I by A4, A6;
hence x in I by A7, BCIALG_1:def 18; :: thesis: verum

end;

0. X in I by BCIALG_1:def 18;
hence I is associative-ideal of X by A5, Def4; :: thesis: verum