let X be BCI-algebra; :: thesis: ( ( for X being BCI-algebra

for x, y being Element of X holds x \ (y \ x) = x ) implies X is BCK-algebra )

assume A1: for X being BCI-algebra

for x, y being Element of X holds x \ (y \ x) = x ; :: thesis: X is BCK-algebra

for z being Element of X holds z ` = 0. X

for x, y being Element of X holds x \ (y \ x) = x ) implies X is BCK-algebra )

assume A1: for X being BCI-algebra

for x, y being Element of X holds x \ (y \ x) = x ; :: thesis: X is BCK-algebra

for z being Element of X holds z ` = 0. X

proof

hence
X is BCK-algebra
by Def8; :: thesis: verum
let z be Element of X; :: thesis: z ` = 0. X

(z \ (0. X)) ` = 0. X by A1;

hence z ` = 0. X by Th2; :: thesis: verum

end;(z \ (0. X)) ` = 0. X by A1;

hence z ` = 0. X by Th2; :: thesis: verum