reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem
  for a being non zero Nat
   ex n being non prime Nat st n divides a|^n-a
  proof
    let a be non zero Nat;
    per cases;
    suppose a is non prime;
      then reconsider n = a as non prime non zero Nat;
      take n;
      a divides a|^n-a|^1 by Th18;
      hence thesis;
    end;
    suppose that
A1:   a is prime and
A2:   a > 2;
      reconsider a as odd Nat by A1,A2,PEPIN:17;
      2*a > 1*a by XREAL_1:68;
      then 2*a > 2*1 by A2,XXREAL_0:2;
      then reconsider n = 2*a as non prime Nat by PEPIN:17;
      take n;
A3:   2 = 2|^1;
A4:   a|^n-a is even;
      a divides a|^n-a|^1 by Th18;
      hence thesis by A3,A4,PEPIN:4,NAT_5:3;
    end;
    suppose that
A5:   a is prime and
A6:   a <= 2;
      a >= 1+1 by A5,NAT_1:13;
      then
A7:   a = 2 by A6,XXREAL_0:1;
A8:   341 = 11*31;
      then reconsider n = 341 as non prime Nat by NAT_4:27,NUMERAL2:24;
      take n;
      thus thesis by A7,A8,Th20,Th21,PEPIN:4,INT_2:30,NAT_4:27,NUMERAL2:24;
    end;
  end;
