#----------------------------------------------------------------
#   Problem:    In a ring, if x*x*x*x = x for all x
#               in the ring, then
#               x*y = y*x for all x,y in the ring.
#
#               Funktionen:     f   : Multiplikation *
#                               J   : Addition +
#                               g   : Inverses
#                               e   : Neutrales Element
#                               a,b : Konstanten
#   Extends luusk 6 to the case x^4
#

j (0,X)       = X.                   # 0 ist a left identity for sum
j (X,0)       = X.                   # 0 ist a right identity for sum
j (g (X),X)   = 0.                   # there exists a left inverse for sum
j (X,g (X))   = 0.                   # there exists a right inverse for sum
j (j (X,Y),Z) = j (X,j (Y,Z)).       # associativity of addition
j (X,Y)       = j(Y,X).              # commutativity of addition
f (f (X,Y),Z) = f (X,f (Y,Z)).       # associativity of multiplication
f (X,j (Y,Z)) = j (f (X,Y),f (X,Z)). # distributivity axioms
f (j (X,Y),Z) = j (f (X,Z),f (Y,Z)). #
f (f (f(X,X),X),X)  = X.             # special hypothese: x*x*x*x = x

f (a,b) != f (b,a).                   # theorem