reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th58:
  for n be prime Nat holds n divides a|^n - a
  proof
    let n be prime Nat;
    defpred P[Nat] means n divides $1|^n - $1;
    L1: P[0] by INT_2:12;
    L2: P[k] implies P[k+1]
    proof
      assume
      A0: P[k];
      n*k divides (k+1)|^n - (k|^n + 1) by Th56; then
      n divides (k+1)|^n - (k|^n + 1) by INT_2:2,INT_1:def 3; then
      n divides k|^n - k + ((k+1)|^n - (k|^n + 1)) by A0,WSIERP_1:4;
      hence thesis;
    end;
    for x be Nat holds P[x] from NAT_1:sch 2(L1,L2);
    hence thesis;
  end;
