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 Th47:
  for PERM be Permutation of Permutations(n), perm st perm is odd
  & for p holds PERM.p = p*perm holds PERM.:{p:p is even} = {q:q is odd}
proof
  set P=Permutations(n);
  let PERM be Permutation of P, perm such that
A1: perm is odd and
A2: for p holds PERM.p=p*perm;
  set E={p:p is even};
  set OD={q:q is odd};
  for y being object holds y in OD iff
   ex x being object st x in dom PERM & x in E & y = PERM.x
  proof
    let y be object;
    thus y in OD implies
      ex x being object st x in dom PERM & x in E & y = PERM.x
    proof
      reconsider perm9=perm" as Element of P by MATRIX_7:18;
A3:   dom PERM=P by FUNCT_2:52;
      n>=2 by A1,Lm3;
      then
A4:   n>=1 by XXREAL_0:2;
      assume y in OD;
      then consider q such that
A5:   y=q and
A6:   q is odd;
A7:   q*(idseq n)=q by MATRIX_1:12;
      perm9 is odd by A1,A4,MATRIX_7:28;
      then
A8:   q*perm9 is even by A6,Th25;
      reconsider qp9=q*perm9 as Element of P by MATRIX_9:39;
      take qp9;
A9:   perm9*perm=idseq n by MATRIX_1:13;
      PERM.qp9=qp9*perm by A2;
      hence thesis by A5,A3,A9,A7,A8,RELAT_1:36;
    end;
    given x being object such that
    x in dom PERM and
A10: x in E and
A11: y = PERM.x;
    consider p such that
A12: p=x and
A13: p is even by A10;
    reconsider pp=p*perm as Element of P by MATRIX_9:39;
A14: PERM.x=p*perm by A2,A12;
    pp is odd by A1,A13,Th25;
    hence thesis by A11,A14;
  end;
  hence thesis by FUNCT_1:def 6;
end;
