let X be BCI-algebra; :: thesis: ( X is p-Semisimple iff for x, y, z being Element of X st x \ y = x \ z holds
y = z )
thus ( X is p-Semisimple implies for x, y, z being Element of X st x \ y = x \ z holds
y = z ) by Lm12; :: thesis: ( ( for x, y, z being Element of X st x \ y = x \ z holds
y = z ) implies X is p-Semisimple )
assume A1: for x, y, z being Element of X st x \ y = x \ z holds
y = z ; :: thesis: X is p-Semisimple
for x being Element of X st x ` = 0. X holds
x = 0. X

proof

let x be Element of X; :: thesis: ( x ` = 0. X implies x = 0. X )
assume x ` = 0. X ; :: thesis: x = 0. X
then x ` = (0. X) ` by Def5;
hence x = 0. X by A1; :: thesis: verum

end;

hence X is p-Semisimple by Th62; :: thesis: verum