reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th49:
  for p being odd Prime st n = (p-1)*(k*p+1) holds 2|^n mod p = 1
  proof
    let p be odd Prime such that
A1: n = (p-1)*(k*p+1);
A2: 1 < p by INT_2:def 4;
    then
A3: p-'1 = p-1 by XREAL_1:233;
A4: 2|^n = 2|^(p-1)|^(k*p+1) by A1,NEWTON:9;
    2|^1,p are_coprime by NAT_5:3;
    then 2|^(p-'1) mod p = 1 by PEPIN:37;
    hence thesis by A2,A3,A4,PEPIN:35;
  end;
