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 holds x \ y = (a \ 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 holds x \ y = (a \ y) \ (a \ x) ) :: thesis: ( ( for a being Element of X st a is being_greatest holds
for x, y being Element of X holds x \ y = (a \ y) \ (a \ x) ) implies X is involutory )
proof
assume A1: X is involutory ; :: thesis: for a being Element of X st a is being_greatest holds
for x, y being Element of X holds x \ y = (a \ y) \ (a \ x)
let a be Element of X; :: thesis: ( a is being_greatest implies for x, y being Element of X holds x \ y = (a \ y) \ (a \ x) )
assume A2: a is being_greatest ; :: thesis: for x, y being Element of X holds x \ y = (a \ y) \ (a \ x)
for x, y being Element of X holds x \ y = (a \ y) \ (a \ x)
proof
let x, y be Element of X; :: thesis: x \ y = (a \ y) \ (a \ x)
x \ y = (a \ (a \ x)) \ y by A1, A2
.= (a \ y) \ (a \ x) by BCIALG_1:7 ;
hence x \ y = (a \ y) \ (a \ x) ; :: thesis: verum
end;
hence for x, y being Element of X holds x \ y = (a \ y) \ (a \ x) ; :: thesis: verum
end;
assume A3: for a being Element of X st a is being_greatest holds
for x, y being Element of X holds x \ y = (a \ 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 A4: a is being_greatest ; :: thesis: for x being Element of X holds a \ (a \ x) = x
now :: thesis: for x being Element of X holds a \ (a \ x) = x
let x be Element of X; :: thesis: a \ (a \ x) = x
(a \ (a \ x)) \ (0. X) = (a \ (0. X)) \ (a \ x) by BCIALG_1:7
.= x \ (0. X) by A3, A4
.= x by BCIALG_1:2 ;
hence a \ (a \ x) = x by BCIALG_1:2; :: thesis: verum
end;
hence for x being Element of X holds a \ (a \ x) = x ; :: thesis: verum