reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th10:
  p is prime & p mod 4 = 1 & n divides p|^k implies n mod 4 = 1
  proof
  assume
A1: p is prime & p mod 4 = 1 & n divides p|^k;
    then consider t be Element of NAT such that
A2: n = p |^ t & t <= k by PEPIN:34;
    defpred P[Nat] means (p |^ $1) mod 4 = 1;
    p|^0 = 1 by NEWTON:4;
    then
A3: P[0] by PEPIN:5;
A4: for i being Nat holds P[i] implies P[i+1]
    proof
      let i be Nat;
      assume P[i];
      then p * (p|^i) mod 4 =1 by A1,Th8;
      hence thesis by NEWTON:6;
    end;
    for i being Nat holds P[i] from NAT_1:sch 2(A3,A4);
    hence thesis by A2;
  end;
