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

proof

let x, y be Element of IT; :: according to BCIALG_4:def 15 :: thesis: x \ (y \ x) = x
thus x \ (y \ x) = x by STRUCT_0:def 10; :: thesis: verum

end;

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