reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem Th36:
  for m,n being Nat holds m > 0 implies (n |^ m) mod n = 0
proof
  let m,n be Nat;
  assume
A1: m > 0;
  defpred P[Nat] means $1 > 0 implies (n |^ $1) mod n = 0;
A2: for m being Nat holds P[m] implies P[m+1]
  proof
    let m be Nat;
    assume P[m];
    P[m+1]
    proof
      reconsider m,n as Element of NAT by ORDINAL1:def 12;
      n*(n |^ m) mod n = ((n mod n)*(n |^ m)) mod n by EULER_2:8
        .= (0*(n |^ m)) mod n by NAT_D:25
        .= 0 by NAT_D:26;
      hence thesis by NEWTON:6;
    end;
    hence thesis;
  end;
A3: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A2);
  hence thesis by A1;
end;
