let X be BCI-algebra; :: thesis: BCK-part X is p-ideal of X
set A = BCK-part X;

A1: now

let x, y, z be Element of X; :: thesis: ( (x \ z) \ (y \ z) in BCK-part X & y in BCK-part X implies x in BCK-part X )
assume that
A2: (x \ z) \ (y \ z) in BCK-part X and
A3: y in BCK-part X ; :: thesis: x in BCK-part X
ex c being Element of X st
( (x \ z) \ (y \ z) = c & 0. X <= c ) by A2;
then ((x \ z) \ (y \ z)) ` = 0. X by BCIALG_1:def 11;
then ((x \ z) ` ) \ ((y \ z) ` ) = 0. X by BCIALG_1:9;
then A4: ((x ` ) \ (z ` )) \ ((y \ z) ` ) = 0. X by BCIALG_1:9;
ex d being Element of X st
( y = d & 0. X <= d ) by A3;
then y ` = 0. X by BCIALG_1:def 11;
then (((x ` ) \ (z ` )) \ ((0. X) \ (z ` ))) \ ((x ` ) \ (0. X)) = ((x ` ) \ (0. X)) ` by A4, BCIALG_1:9;
then (((x ` ) \ (z ` )) \ ((0. X) \ (z ` ))) \ ((x ` ) \ (0. X)) = (x ` ) ` by BCIALG_1:2;
then 0. X = (0. X) \ ((0. X) \ x) by BCIALG_1:def 3;
then (0. X) \ x = 0. X by BCIALG_1:1;
then 0. X <= x by BCIALG_1:def 11;
hence x in BCK-part X ; :: thesis: verum

end;

(0. X) \ (0. X) = 0. X by BCIALG_1:def 5;
then 0. X <= 0. X by BCIALG_1:def 11;
then 0. X in BCK-part X ;
hence BCK-part X is p-ideal of X by A1, Def5; :: thesis: verum