reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem
  for n being odd Nat holds n divides 2|^(n!) - 1
  proof
    let n be odd Nat;
    set E = Euler n;
    per cases;
    suppose n <= 1;
      then n = 1 by NAT_1:25;
      hence thesis by NAT_D:6;
    end;
    suppose n > 1;
      then (2|^E) mod n = 1 by Lm2,NAT_5:3,EULER_2:18;
      then
A1:   n divides 2|^E-1 by PEPIN:8;
      2|^E-1 divides 2|^(n!)-1 by Th47,NUMBER05:19;
      hence n divides 2|^(n!)-1 by A1,INT_2:9;
    end;
  end;
