let X be BCI-algebra; :: thesis: for x being Element of X holds
( x in AtomSet X iff for z being Element of X holds (z `) \ (x `) = x \ z )
let x be Element of X; :: thesis: ( x in AtomSet X iff for z being Element of X holds (z `) \ (x `) = x \ z )
thus ( x in AtomSet X implies for z being Element of X holds (z `) \ (x `) = x \ z ) :: thesis: ( ( for z being Element of X holds (z `) \ (x `) = x \ z ) implies x in AtomSet X )

proof

assume x in AtomSet X ; :: thesis: for z being Element of X holds (z `) \ (x `) = x \ z
then A1: ex x1 being Element of X st
( x = x1 & x1 is atom ) ;
let z be Element of X; :: thesis: (z `) \ (x `) = x \ z
(z \ (z \ x)) \ x = 0. X by Th1;
then z \ (z \ x) = x by A1;
then x \ z = (z \ z) \ (z \ x) by Th7;
then x \ z = (z \ x) ` by Def5;
hence (z `) \ (x `) = x \ z by Th9; :: thesis: verum

end;

assume A2: for z being Element of X holds (z `) \ (x `) = x \ z ; :: thesis: x in AtomSet X

now :: thesis: for z being Element of X st z \ x = 0. X holds
z = x

let z be Element of X; :: thesis: ( z \ x = 0. X implies z = x )
assume A3: z \ x = 0. X ; :: thesis: z = x
then (z \ x) ` = 0. X by Def5;
then (z `) \ (x `) = 0. X by Th9;
then x \ z = 0. X by A2;
hence z = x by A3, Def7; :: thesis: verum

end;

then x is atom ;
hence x in AtomSet X ; :: thesis: verum