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 ((x \ z) `) ` = x \ z )
let x be Element of X; :: thesis: ( x in AtomSet X iff for z being Element of X holds ((x \ z) `) ` = x \ z )
thus ( x in AtomSet X implies for z being Element of X holds ((x \ z) `) ` = x \ z ) :: thesis: ( ( for z being Element of X holds ((x \ z) `) ` = x \ z ) implies x in AtomSet X )
proof
assume A1: x in AtomSet X ; :: thesis: for z being Element of X holds ((x \ z) `) ` = x \ z
let z be Element of X; :: thesis: ((x \ z) `) ` = x \ z
A2: (z \ (z \ x)) \ x = 0. X by Th1;
ex x1 being Element of X st
( x = x1 & x1 is atom ) by A1;
then z \ (z \ x) = x by A2;
then ((x \ z) `) ` = (((z \ z) \ (z \ x)) `) ` by Th7;
then ((x \ z) `) ` = (((z \ x) `) `) ` by Def5;
then ((x \ z) `) ` = (z \ x) ` by Th8;
hence ((x \ z) `) ` = x \ z by A1, Th30; :: thesis: verum
end;
assume for z being Element of X holds ((x \ z) `) ` = x \ z ; :: thesis: x in AtomSet X
then ((x \ (0. X)) `) ` = x \ (0. X) ;
then (x `) ` = x \ (0. X) by Th2;
then (x `) ` = x by Th2;
hence x in AtomSet X by Th29; :: thesis: verum