
theorem lemmandiv:
  for n,k being Nat st n is odd holds n divides k|^n + (n-k)|^n
  proof
    let n,k be Nat;
    assume no: n is odd;
    then reconsider p=n as positive Nat;
    kk: k|^p + (p+(-k))|^p = k|^p + Sum((p,-k) In_Power p) by NEWTON:30
    .= k|^p + (p|^p + (-k)|^p + Sum((((p,-k) In_Power p)|p)/^1)) by lemman02
    .= k|^p + (p|^p + - k|^p + Sum((((p,-k) In_Power p)|p)/^1)) by no,POWER:2
    .= p|^p + Sum((((p,-k) In_Power p)|p)/^1);
    pp: p divides p|^p by NAT_3:3;
    reconsider S=Sum((((p,-k) In_Power p)|p)/^1) as Integer;
    reconsider f=(((p,-k) In_Power p)|p)/^1 as INT-valued FinSequence;
    now
      let o be Nat;
      assume o: o in dom f;
      then f <> {};
      then 1<=len (((p,-k) In_Power p)|p) by RFINSEQ:def 1;
      then rf: len f = len (((p,-k) In_Power p)|p)-1 &
      for l be Nat st l in dom f holds f.l = (((p,-k) In_Power p)|p).(l+1)
      by RFINSEQ:def 1;

      elen: len ((p,-k) In_Power p) = n+1 &
      for i,l,m being Nat st i in dom ((p,-k) In_Power p) &
      m = i - 1 & l = n-m holds
      ((p,-k) In_Power p).i = (n choose m) * p|^l * (-k)|^m by NEWTON:def 4;
      x: 0+1<=o+1 by XREAL_1:6;
      pz: p+0<=n+1 by XREAL_1:6;
      len (((p,-k) In_Power p)|p) = p by elen,pz,FINSEQ_1:17;
      then lp: len f = p-1 by rf;
      o in Seg len f by o,FINSEQ_1:def 3;
      then o <= p-1 by lp,FINSEQ_1:1;
      then op: o+1 <= p-1+1 by XREAL_1:6;
      then el: o+1 in Seg p by x;
      set i=o+1;
      i<p+1 by NAT_1:13,op;
      then i in Seg len ((p,-k) In_Power p) by x,elen;
      then i: i in dom ((p,-k) In_Power p) by FINSEQ_1:def 3;
      reconsider m=i-1 as Nat;
      po: p-o >= o+1-o by op,XREAL_1:9;
      then reconsider l=n-m as Nat;
      fo: f.o = (((p,-k) In_Power p)|p).(o+1) by o,rf
      .= ((p,-k) In_Power p).(o+1) by el,FUNCT_1:49
      .= (n choose m) * p|^l * (-k)|^m by elen,i
      .= p|^l * ((n choose m) * (-k)|^m);
      p divides p|^l by po,NAT_3:3;
      hence n divides f.o by fo,INT_2:2;
    end;
    then n divides Sum f by NEWTON04:80;
    then n divides k|^n + (n+(-k))|^n by kk,pp,WSIERP_1:4;
    hence n divides k|^n + (n-k)|^n;
  end;
