let X be BCI-algebra; :: thesis: ( X is quasi-associative iff for x being Element of X holds x ` <= x )
thus ( X is quasi-associative implies for x being Element of X holds x ` <= x ) :: thesis: ( ( for x being Element of X holds x ` <= x ) implies X is quasi-associative )

proof

assume A1: X is quasi-associative ; :: thesis: for x being Element of X holds x ` <= x
let x be Element of X; :: thesis: x ` <= x
((x `) `) \ x = 0. X by Th1;
then (x `) \ x = 0. X by A1;
hence x ` <= x ; :: thesis: verum

end;

assume for x being Element of X holds x ` <= x ; :: thesis: X is quasi-associative
then for x, y being Element of X holds (x \ y) ` = (y \ x) ` by Lm15;
then for x, y being Element of X holds (x `) \ y = (x \ y) ` by Lm16;
then for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X by Lm17;
hence X is quasi-associative by Lm18; :: thesis: verum