
theorem
  for p be Sophie_Germain Prime st p > 2 & p mod 4 = 1 holds
  ex q be Safe Prime st Mersenne(p) mod q = q-2
proof
  let p be Sophie_Germain Prime;
  assume that
A1: p > 2 and
A2: p mod 4 = 1;
  set q = 2*p+1;
A3: q is Safe Prime by Def1,Def2;
A4: q > 5 by A1,Lm1;
  then
A5: q > 5-3 by XREAL_1:51;
  then 2,q are_coprime by A3,INT_2:28,30;
  then
A6: 2 gcd q = 1 by INT_2:def 3;
  p = (p div 4)*4+1 by A2,INT_1:59;
  then q = (p div 4)*8+3;
  then q mod 8 = 3 mod 8 by NAT_D:21
    .= 3 by NAT_D:24;
  then not 2 is_quadratic_residue_mod q by A3,A5,INT_5:44;
  then 2|^((2*p+1-'1) div 2) mod q = q-1 by A3,A5,A6,INT_5:19;
  then
A7: 2|^((2*p) div 2) mod q = q-1 by NAT_D:34;
A8: q > 5-4 by A4,XREAL_1:51;
  then q >= 1+1 by NAT_1:13;
  then
A9: q-2 is Nat by NAT_1:21;
  Mersenne(p) mod q = ((2|^p mod q)-(1 mod q)) mod q by INT_6:7
    .= ((q-1)-(1 mod q)) mod q by A7,NAT_D:18
    .= ((q-1)-1) mod q by A8,PEPIN:5
    .= q-2 by A9,NAT_D:24,XREAL_1:44;
  hence thesis by A3;
end;
