let X be BCK-algebra; :: thesis: for x, y being Element of X holds
( x \ (x \ y) <= y & x \ (x \ y) <= x )
let x, y be Element of X; :: thesis: ( x \ (x \ y) <= y & x \ (x \ y) <= x )
A1: (0. X) \ (x \ y) = (x \ y) `
.= 0. X by BCIALG_1:def 8 ;
((x \ (x \ y)) \ ((0. X) \ (x \ y))) \ (x \ (0. X)) = 0. X by BCIALG_1:def 3;
then ((x \ (x \ y)) \ (0. X)) \ x = 0. X by A1, BCIALG_1:2;
then ( (x \ (x \ y)) \ y = 0. X & (x \ (x \ y)) \ x = 0. X ) by BCIALG_1:1, BCIALG_1:2;
hence ( x \ (x \ y) <= y & x \ (x \ y) <= x ) ; :: thesis: verum