reserve x, y for object, X for set,
  i, j, k, l, n, m for Nat,
  D for non empty set,
  K for commutative Ring,
  a,b for Element of K,
  perm, p, q for Element of Permutations(n),
  Perm,P for Permutation of Seg n,
  F for Function of Seg n,Seg n,
  perm2, p2, q2, pq2 for Element of Permutations(n+2),
  Perm2 for Permutation of Seg (n+2);
reserve s for Element of 2Set Seg (n+2);
reserve pD for FinSequence of D,
  M for Matrix of n,m,D,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem
  for n st n >= 2 ex ODD,EVEN be finite set st EVEN ={p:p is even} & ODD
= {q:q is odd} & EVEN /\ ODD = {} & EVEN \/ ODD =Permutations(n) & card EVEN =
  card ODD
proof
  let n such that
A1: n >= 2;
  1 <= n by A1,XXREAL_0:2;
  then
A2: 1 in Seg n;
  2 in Seg n by A1;
  then consider O,E be finite set such that
A3: E ={p:p is even} & O = {q:q is odd} and
A4: E /\ O = {} & E \/ O = Permutations(n) and
A5: ex P be Function of E,O, perm st perm is being_transposition & perm.
  1=2 & dom P=E & P is bijective & for p st p in E holds P.p=p*perm by A2,Lm8;
  consider P be Function of E,O, perm such that
  perm is being_transposition and
  perm.1=2 and
A6: dom P=E and
A7: P is bijective and
  for p st p in E holds P.p=p*perm by A5;
  rng P=O by A7,FUNCT_2:def 3;
  then E,O are_equipotent by A6,A7,WELLORD2:def 4;
  then card E=card O by CARD_1:5;
  hence thesis by A3,A4;
end;
