reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th14:
  for n being even Nat st n divides 2|^n+2
   ex k being non zero odd Nat st 2|^n+2 = n*k
  proof
    let n be even Nat;
    assume
A1: n divides 2|^n+2;
    then consider k being Nat such that
A2: 2|^n+2 = n*k by NAT_D:def 3;
    reconsider n as non zero Nat by A1;
    now
      assume k is even;
      then consider a being Nat such that
A3:   k = 2*a;
A4:   2|^n+2 = 2*(a*n) by A2,A3;
      2|^(n-1+1)+2 = 2*2|^(n-1)+2 by NEWTON:6
      .= 2*(2|^(n-1)+1);
      hence contradiction by A4,XCMPLX_1:5;
    end;
    then reconsider k as non zero odd Nat;
    take k;
    thus thesis by A2;
  end;
