let X be BCI-algebra; :: thesis: ( X is p-Semisimple iff for x, z being Element of X holds ((x \ z) `) ` = x \ z )
thus ( X is p-Semisimple implies for x, z being Element of X holds ((x \ z) `) ` = x \ z ) ; :: thesis: ( ( for x, z being Element of X holds ((x \ z) `) ` = x \ z ) implies X is p-Semisimple )
assume A1: for x, z being Element of X holds ((x \ z) `) ` = x \ z ; :: 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 `) ` = x \ (0. X) by Th2;
hence (x `) ` = x by Th2; :: thesis: verum
end;
hence X is p-Semisimple by Th54; :: thesis: verum