let X be BCI-algebra; :: thesis: for x being Element of X holds

( x in AtomSet X iff for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u )

let x be Element of X; :: thesis: ( x in AtomSet X iff for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u )

thus ( x in AtomSet X implies for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u ) :: thesis: ( ( for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u ) implies x in AtomSet X )

hence x in AtomSet X ; :: thesis: verum

( x in AtomSet X iff for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u )

let x be Element of X; :: thesis: ( x in AtomSet X iff for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u )

thus ( x in AtomSet X implies for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u ) :: thesis: ( ( for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u ) implies x in AtomSet X )

proof

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

then A1: ex x1 being Element of X st

( x = x1 & x1 is atom ) ;

let u, z be Element of X; :: thesis: (z \ u) \ (z \ x) = x \ u

(z \ (z \ x)) \ x = 0. X by Th1;

then z \ (z \ x) = x by A1;

hence (z \ u) \ (z \ x) = x \ u by Th7; :: thesis: verum

end;then A1: ex x1 being Element of X st

( x = x1 & x1 is atom ) ;

let u, z be Element of X; :: thesis: (z \ u) \ (z \ x) = x \ u

(z \ (z \ x)) \ x = 0. X by Th1;

then z \ (z \ x) = x by A1;

hence (z \ u) \ (z \ x) = x \ u by Th7; :: thesis: verum

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

z = x

then
x is atom
;z = x

let z be Element of X; :: thesis: ( z \ x = 0. X implies z = x )

assume z \ x = 0. X ; :: thesis: z = x

then (z \ (0. X)) \ (0. X) = x \ (0. X) by A2;

then (z \ (0. X)) \ (0. X) = x by Th2;

then z \ (0. X) = x by Th2;

hence z = x by Th2; :: thesis: verum

end;assume z \ x = 0. X ; :: thesis: z = x

then (z \ (0. X)) \ (0. X) = x \ (0. X) by A2;

then (z \ (0. X)) \ (0. X) = x by Th2;

then z \ (0. X) = x by Th2;

hence z = x by Th2; :: thesis: verum

hence x in AtomSet X ; :: thesis: verum