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

proof

let x, y be Element of X; :: thesis: Polynom (0,0,x,y) = Polynom (0,n,y,x)
A3: Polynom (i,j,x,y) = Polynom (m,n,y,x) by Def3;
A4: i + 1 >= j + 1 by A1, XREAL_1:6;
( (x,(x \ y)) to_power (j + 1) = (x,(x \ y)) to_power (m + 1) & j + 1 < m + 1 ) by A1, A2, Th20, XREAL_1:6;
then (x,(x \ y)) to_power (i + 1) = (x,(x \ y)) to_power (0 + 1) by A1, A4, Th6;
hence Polynom (0,0,x,y) = Polynom (0,n,y,x) by A1, A3, Th20; :: thesis: verum

end;

then reconsider X = X as BCK-algebra of 0 , 0 , 0 ,n by Def3;
A5: for x, y being Element of X holds Polynom (0,0,x,y) <= Polynom (0,0,y,x)

proof

let x, y be Element of X; :: thesis: Polynom (0,0,x,y) <= Polynom (0,0,y,x)
Polynom (0,0,x,y) = Polynom (0,n,y,x) by Def3;
hence Polynom (0,0,x,y) <= Polynom (0,0,y,x) by Th5; :: thesis: verum

end;

for x, y being Element of X holds Polynom (0,0,y,x) = Polynom (0,0,x,y)

proof

let x, y be Element of X; :: thesis: Polynom (0,0,y,x) = Polynom (0,0,x,y)
Polynom (0,0,x,y) <= Polynom (0,0,y,x) by A5;
then A6: (Polynom (0,0,x,y)) \ (Polynom (0,0,y,x)) = 0. X ;
Polynom (0,0,y,x) <= Polynom (0,0,x,y) by A5;
then (Polynom (0,0,y,x)) \ (Polynom (0,0,x,y)) = 0. X ;
hence Polynom (0,0,y,x) = Polynom (0,0,x,y) by A6, BCIALG_1:def 7; :: thesis: verum

end;

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