let i, j, m, n be Element of NAT ; :: thesis: for X being BCK-algebra of i,j,m,n st i = min i,j,m,n & i = j holds
X is BCK-algebra of i,i,i,i
let X be BCK-algebra of i,j,m,n; :: thesis: ( i = min i,j,m,n & i = j implies X is BCK-algebra of i,i,i,i )
assume A1: i = min i,j,m,n ; :: thesis: ( not i = j or X is BCK-algebra of i,i,i,i )
assume A2: i = j ; :: thesis: X is BCK-algebra of i,i,i,i
A3: for x, y being Element of X holds Polynom i,i,x,y <= Polynom i,i,y,x

proof

let x, y be Element of X; :: thesis: Polynom i,i,x,y <= Polynom i,i,y,x
i <= n by A1, Th25;
then A4: (y,(y \ x) to_power (i + 1)),(x \ y) to_power n <= (y,(y \ x) to_power (i + 1)),(x \ y) to_power i by Th5;
i <= m by A1, Th25;
then i + 1 <= m + 1 by XREAL_1:8;
then A5: (y,(y \ x) to_power (m + 1)),(x \ y) to_power n <= (y,(y \ x) to_power (i + 1)),(x \ y) to_power n by Th5, BCIALG_2:19;
Polynom i,j,x,y = Polynom m,n,y,x by Def3;
hence Polynom i,i,x,y <= Polynom i,i,y,x by A2, A5, A4, Th1; :: thesis: verum

end;

for x, y being Element of X holds Polynom i,i,y,x = Polynom i,i,x,y

proof

let x, y be Element of X; :: thesis: Polynom i,i,y,x = Polynom i,i,x,y
Polynom i,i,x,y <= Polynom i,i,y,x by A3;
then A6: (Polynom i,i,x,y) \ (Polynom i,i,y,x) = 0. X by BCIALG_1:def 11;
Polynom i,i,y,x <= Polynom i,i,x,y by A3;
then (Polynom i,i,y,x) \ (Polynom i,i,x,y) = 0. X by BCIALG_1:def 11;
hence Polynom i,i,y,x = Polynom i,i,x,y by A6, BCIALG_1:def 7; :: thesis: verum

end;

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