let i, j, k, l, m, n be Nat; :: thesis: for X being BCK-algebra of i,j,m,n st l >= j & k >= n holds
X is BCK-algebra of k,l,l,k
let X be BCK-algebra of i,j,m,n; :: thesis: ( l >= j & k >= n implies X is BCK-algebra of k,l,l,k )
assume that
A1: l >= j and
A2: k >= n ; :: thesis: X is BCK-algebra of k,l,l,k
l - j is Element of NAT by A1, NAT_1:21;
then consider t being Element of NAT such that
A3: t = l - j ;
k - n is Element of NAT by A2, NAT_1:21;
then consider t1 being Element of NAT such that
A4: t1 = k - n ;
X is BCK-algebra of n,j,m,n by Th22;
then X is BCK-algebra of n,j,j,n by Th21;
then X is BCK-algebra of n + t1,j,j,n + t1 by Th17;
then X is BCK-algebra of k,j + t,j + t,k by A4, Th18;
hence X is BCK-algebra of k,l,l,k by A3; :: thesis: verum