let X be BCI-algebra; :: thesis: ( X is quasi-associative iff for x, y being Element of X holds (x \ y) ` = (y \ x) ` )

thus ( X is quasi-associative implies for x, y being Element of X holds (x \ y) ` = (y \ x) ` ) :: thesis: ( ( for x, y being Element of X holds (x \ y) ` = (y \ x) ` ) implies X is quasi-associative )

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

thus ( X is quasi-associative implies for x, y being Element of X holds (x \ y) ` = (y \ x) ` ) :: thesis: ( ( for x, y being Element of X holds (x \ y) ` = (y \ x) ` ) implies X is quasi-associative )

proof

assume
for x, y being Element of X holds (x \ y) ` = (y \ x) `
; :: thesis: X is quasi-associative
assume
X is quasi-associative
; :: thesis: for x, y being Element of X holds (x \ y) ` = (y \ x) `

then for x being Element of X holds x ` <= x by Th71;

hence for x, y being Element of X holds (x \ y) ` = (y \ x) ` by Lm15; :: thesis: verum

end;then for x being Element of X holds x ` <= x by Th71;

hence for x, y being Element of X holds (x \ y) ` = (y \ x) ` by Lm15; :: thesis: verum

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