let i, j, m, n be Element of NAT ; :: thesis: for X being BCK-algebra of i,j,m,n st m = 0 & j > 0 holds
X is BCK-algebra of 0 ,1, 0 ,1
let X be BCK-algebra of i,j,m,n; :: thesis: ( m = 0 & j > 0 implies X is BCK-algebra of 0 ,1, 0 ,1 )
reconsider X = X as BCK-algebra of i + 1,j,m,n + 1 by Th17;
assume that
A1: m = 0 and
A2: j > 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: (i + 1) + 1 > (m + 1) + 0 by A1, XREAL_1:10;
A4: (x,(x \ y) to_power (0 + 1)),(y \ x) to_power (j + 1) = (x,(x \ y) to_power (0 + 1)),(y \ x) to_power (m + 1) by Th20;
A5: j + 1 > m + 1 by A1, A2, XREAL_1:8;
( n + 1 >= m + 1 & (y,(y \ x) to_power (0 + 1)),(x \ y) to_power (j + 1) = (y,(y \ x) to_power (0 + 1)),(x \ y) to_power (m + 1) ) by A1, Th20, XREAL_1:8;
then A6: (y,(y \ x) to_power (0 + 1)),(x \ y) to_power (0 + 1) = (y,(y \ x) to_power (0 + 1)),(x \ y) to_power (n + 1) by A1, A5, Th6;
( Polynom (i + 1),j,x,y = Polynom m,(n + 1),y,x & x,(x \ y) to_power (j + 1) = x,(x \ y) to_power (m + 1) ) by Def3, Th20;
then (x,(x \ y) to_power (0 + 1)),(y \ x) to_power j = (y,(y \ x) to_power (0 + 1)),(x \ y) to_power (n + 1) 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