reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;
reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;

theorem
  for X being non empty set
  for x being Tuple of 4,X holds
  pi_1324(pi_1243(x)) = pi_1423(x) & pi_2143(pi_1243(x)) = pi_2134(x) &
  pi_3412(pi_1243(x)) = pi_4312(x) & pi_4321(pi_1243(x)) = pi_3421(x) &
  pi_3412(pi_1324(x)) = pi_2413(x) & pi_2143(pi_1324(x)) = pi_3142(x) &
  pi_4321(pi_1324(x)) = pi_4231(x) & pi_3412(pi_1423(x)) = pi_2314(x) &
  pi_2143(pi_1423(x)) = pi_4132(x) & pi_4321(pi_1423(x)) = pi_3241(x) &
  pi_1243(pi_1423(x)) = pi_1432(x) & pi_4321(pi_1432(x)) = pi_2341(x) &
  pi_3412(pi_1432(x)) = pi_3214(x) & pi_2143(pi_1432(x)) = pi_4123(x) &
  pi_4321(pi_3124(x)) = pi_4213(x) & pi_3412(pi_3124(x)) = pi_2431(x) &
  pi_2143(pi_3124(x)) = pi_1342(x) & pi_4312(pi_3124(x)) = pi_4231(x) &
  pi_4321(pi_3124(x)) = pi_4213(x);
