reconsider IT = BCI_S-EXAMPLE as BCK-Algebra_with_Condition(S) ;
IT is positive-implicative

proof

let x, y be Element of IT; :: according to BCIALG_4:def 11 :: thesis: (x \ y) \ y = x \ y
thus (x \ y) \ y = x \ y by STRUCT_0:def 10; :: thesis: verum

end;

hence ex b₁ being BCK-Algebra_with_Condition(S) st b₁ is positive-implicative ; :: thesis: verum