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

proof

assume x in AtomSet X ; :: thesis: (x `) ` = x
then ((0. X) `) \ (x `) = x \ (0. X) by Th28;
then ((0. X) `) \ (x `) = x by Th2;
hence (x `) ` = x by Def5; :: thesis: verum

end;

assume A1: (x `) ` = x ; :: 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 A2: z \ x = 0. X ; :: thesis: z = x
then ((z \ x) \ (x `)) \ (z \ (0. X)) = x \ z by A1, Th2;
then 0. X = x \ z by Def3;
hence z = x by A2, Def7; :: thesis: verum

end;

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