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

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

assume A1: for X being BCI-algebra

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

assume A1: for X being BCI-algebra

for x, y being Element of X holds (x \ y) \ y = x \ y ; :: 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 `) \ ((z `) \ z) = (z `) \ (z `) by A1;

then (z `) \ ((z `) \ z) = 0. X by Def5;

hence z ` = 0. X by Th1; :: thesis: verum

end;(z `) \ ((z `) \ z) = (z `) \ (z `) by A1;

then (z `) \ ((z `) \ z) = 0. X by Def5;

hence z ` = 0. X by Th1; :: thesis: verum