let X be bounded BCK-algebra; :: thesis: ( X is involutory iff for a being Element of X st a is being_greatest holds
for x, y being Element of X st x <= a \ y holds
y <= a \ x )
thus ( X is involutory implies for a being Element of X st a is being_greatest holds
for x, y being Element of X st x <= a \ y holds
y <= a \ x ) by Th23; :: thesis: ( ( for a being Element of X st a is being_greatest holds
for x, y being Element of X st x <= a \ y holds
y <= a \ x ) implies X is involutory )
assume A1: for a being Element of X st a is being_greatest holds
for x, y being Element of X st x <= a \ y holds
y <= a \ x ; :: thesis: X is involutory
let a be Element of X; :: according to BCIALG_3:def 7 :: thesis: ( a is being_greatest implies for x being Element of X holds a \ (a \ x) = x )
assume A2: a is being_greatest ; :: thesis: for x being Element of X holds a \ (a \ x) = x
let x be Element of X; :: thesis: a \ (a \ x) = x
(a \ x) \ (a \ x) = 0. X by BCIALG_1:def 5;
then a \ x <= a \ x ;
then x <= a \ (a \ x) by A1, A2;
then A3: x \ (a \ (a \ x)) = 0. X ;
(a \ (a \ x)) \ x = (a \ x) \ (a \ x) by BCIALG_1:7
.= 0. X by BCIALG_1:def 5 ;
hence a \ (a \ x) = x by A3, BCIALG_1:def 7; :: thesis: verum