let X be BCI-algebra; :: thesis: ( X is p-Semisimple iff for x, y, z being Element of X st y \ x = z \ x holds
y = z )
thus ( X is p-Semisimple implies for x, y, z being Element of X st y \ x = z \ x holds
y = z ) by Lm14; :: thesis: ( ( for x, y, z being Element of X st y \ x = z \ x holds
y = z ) implies X is p-Semisimple )
assume A1: for x, y, z being Element of X st y \ x = z \ x holds
y = z ; :: thesis: X is p-Semisimple
for x, y being Element of X holds x \ (x \ y) = y

proof

let x, y be Element of X; :: thesis: x \ (x \ y) = y
(x \ (x \ y)) \ y = 0. X by Th1;
then (x \ (x \ y)) \ y = y \ y by Def5;
hence x \ (x \ y) = y by A1; :: thesis: verum

end;

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