let X be BCI-algebra; :: thesis: ( X is implicative BCI-algebra implies X is BCI-algebra of 0 ,1, 0 , 0 )
assume A1: X is implicative BCI-algebra ; :: thesis: X is BCI-algebra of 0 ,1, 0 , 0
for x, y being Element of X holds Polynom 0 ,1,x,y = Polynom 0 ,0 ,y,x

proof

let x, y be Element of X; :: thesis: Polynom 0 ,1,x,y = Polynom 0 ,0 ,y,x
A2: (x \ (x \ y)) \ (y \ x) = y \ (y \ x) by A1, BCIALG_1:def 24;
(x,(x \ y) to_power 1),(y \ x) to_power 1 = (x \ (x \ y)),(y \ x) to_power 1 by BCIALG_2:2
.= (x \ (x \ y)) \ (y \ x) by BCIALG_2:2
.= y,(y \ x) to_power 1 by A2, BCIALG_2:2
.= (y,(y \ x) to_power 1),(x \ y) to_power 0 by BCIALG_2:1 ;
hence Polynom 0 ,1,x,y = Polynom 0 ,0 ,y,x ; :: thesis: verum

end;

hence X is BCI-algebra of 0 ,1, 0 , 0 by Def3; :: thesis: verum