let X be BCI-algebra; :: thesis: ( X is associative BCI-algebra implies ( X is BCI-algebra of 0 ,1, 0 , 0 & X is BCI-algebra of 1, 0 , 0 , 0 ) )
assume A1: X is associative BCI-algebra ; :: thesis: ( X is BCI-algebra of 0 ,1, 0 , 0 & X is BCI-algebra of 1, 0 , 0 , 0 )
for x being Element of X holds (x `) ` = x
proof
let x be Element of X; :: thesis: (x `) ` = x
x ` = x by ;
hence (x `) ` = x ; :: thesis: verum
end;
then A2: X is p-Semisimple by BCIALG_1:54;
for x, y being Element of X holds Polynom (1,0,x,y) = Polynom (0,0,y,x)
proof
let x, y be Element of X; :: thesis: Polynom (1,0,x,y) = Polynom (0,0,y,x)
x \ (x \ y) = y by A2;
then A3: (x \ (x \ y)) \ (x \ y) = y \ (y \ x) by ;
(((x,(x \ y)) to_power (1 + 1)),(y \ x)) to_power 0 = (x,(x \ y)) to_power 2 by BCIALG_2:1
.= (x \ (x \ y)) \ (x \ y) by BCIALG_2:3
.= (y,(y \ x)) to_power 1 by
.= (((y,(y \ x)) to_power 1),(x \ y)) to_power 0 by BCIALG_2:1 ;
hence Polynom (1,0,x,y) = Polynom (0,0,y,x) ; :: thesis: verum
end;
hence ( X is BCI-algebra of 0 ,1, 0 , 0 & X is BCI-algebra of 1, 0 , 0 , 0 ) by ; :: thesis: verum