let i, j, m, n be Element of NAT ; :: thesis: for X being BCK-algebra of i,j,m,n st i = 0 & n <> 0 holds
X is BCK-algebra of 0 ,1, 0 ,1
let X be BCK-algebra of i,j,m,n; :: thesis: ( i = 0 & n <> 0 implies X is BCK-algebra of 0 ,1, 0 ,1 )
reconsider X = X as BCK-algebra of i,j + 1,m + 1,n by Th18;
assume that
A1: i = 0 and
A2: n <> 0 ; :: thesis: X is BCK-algebra of 0 ,1, 0 ,1
for x, y being Element of X holds Polynom 0 ,1,x,y = Polynom 0 ,1,y,x

proof

let x, y be Element of X; :: thesis: Polynom 0 ,1,x,y = Polynom 0 ,1,y,x
A3: (m + 1) + 1 > (i + 1) + 0 by A1, XREAL_1:10;
A4: (y,(y \ x) to_power (0 + 1)),(x \ y) to_power (i + 1) = (y,(y \ x) to_power (0 + 1)),(x \ y) to_power (n + 1) by Th19;
A5: n + 1 > i + 1 by A1, A2, XREAL_1:8;
( j + 1 >= i + 1 & (x,(x \ y) to_power (0 + 1)),(y \ x) to_power (n + 1) = (x,(x \ y) to_power (0 + 1)),(y \ x) to_power (i + 1) ) by A1, Th19, XREAL_1:8;
then A6: (x,(x \ y) to_power (0 + 1)),(y \ x) to_power (0 + 1) = (x,(x \ y) to_power (0 + 1)),(y \ x) to_power (j + 1) by A1, A5, Th6;
( Polynom i,(j + 1),x,y = Polynom (m + 1),n,y,x & y,(y \ x) to_power (n + 1) = y,(y \ x) to_power (i + 1) ) by Def3, Th19;
then (x,(x \ y) to_power (0 + 1)),(y \ x) to_power (j + 1) = (y,(y \ x) to_power (0 + 1)),(x \ y) to_power n by A1, A5, A3, Th6;
hence Polynom 0 ,1,x,y = Polynom 0 ,1,y,x by A1, A5, A6, A4, Th6, NAT_1:14; :: thesis: verum

end;

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