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

now :: thesis: for x being Element of X holds (x `) ` = x

let x be Element of X; :: thesis: (x `) ` = x
((x \ (0. X)) `) ` = x \ (0. X) by A1;
then ((x \ (0. X)) `) ` = x by Th2;
hence (x `) ` = x by Th2; :: thesis: verum

end;

hence X is p-Semisimple by Th54; :: thesis: verum