reserve x,y for object,
        i,j,k,m,n for Nat;

theorem
  card the set of all p where p is odd-valued a_partition of n
    =
  card the set of all p where p is one-to-one a_partition of n
proof
  set OddVall = the set of all p where p is odd-valued a_partition of n;
  set OneToOne = the set of all p where p is one-to-one a_partition of n;
  A1:the odd-valued a_partition of n in OddVall;
  ex E be Function of OneToOne, OddVall st
    for p be one-to-one a_partition of n holds E.p = Euler_transformation p
  proof
    defpred P[object,object] means
      for p be one-to-one a_partition of n st p=$1 holds
        $2=Euler_transformation p;
    A2:for x be object st x in OneToOne ex y be object st y in OddVall & P[x,y]
    proof
      let x be object;
      assume x in OneToOne;
      then ex q be one-to-one a_partition of n st x=q & not contradiction;
      then reconsider x as one-to-one a_partition of n;
      take Euler_transformation x;
      thus thesis;
    end;
    consider f be Function of OneToOne, OddVall such that
      A3: for x be object st x in OneToOne holds P[x,f.x]
      from FUNCT_2:sch 1(A2);
    take f;
    let p be one-to-one a_partition of n;
    p in OneToOne;
    hence thesis by A3;
  end;
  then consider E be Function of OneToOne, OddVall such that
    A4:for p be one-to-one a_partition of n
    holds E.p=Euler_transformation p;
  A5:dom E = OneToOne by A1,FUNCT_2:def 1;
  OddVall c= rng E
  proof
    let p be object;
    assume p in OddVall;
    then ex o be odd-valued a_partition of n st o=p & not contradiction;
    then reconsider p as odd-valued a_partition of n;
    consider q be one-to-one a_partition of n such that
    A6: p = Euler_transformation q by Th15;
    q in dom E & p=E.q by A6, A4,A5;
    hence thesis by FUNCT_1:def 3;
  end;
  then A7: rng E = OddVall;
  E is one-to-one
  proof
    let p1,p2 be object;
    assume A8:p1 in dom E & p2 in dom E & E.p1=E.p2;
    then
     (ex o be one-to-one a_partition of n st o=p1 & not contradiction) &
     ex o be one-to-one a_partition of n st o=p2 & not contradiction by A5;
    then reconsider p1,p2 as one-to-one a_partition of n;
    E.p1 = Euler_transformation p1 & E.p2 = Euler_transformation p2 by A4;
    hence thesis by Th14,A8;
  end;
  hence thesis by A5,A7,CARD_1:70;
end;
