let X be BCI-algebra; :: thesis: ( X is quasi-associative iff for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X )
thus ( X is quasi-associative implies for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X ) :: thesis: ( ( for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X ) implies X is quasi-associative )

proof

assume X is quasi-associative ; :: thesis: for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X
then for x, y being Element of X holds (x `) \ y = (x \ y) ` by Th73;
hence for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X by Lm17; :: thesis: verum

end;

thus ( ( for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X ) implies X is quasi-associative ) by Lm18; :: thesis: verum