
theorem
  for p be Sophie_Germain Prime st p > 2 & p mod 4 = 3
  ex q be Safe Prime st q divides Mersenne(p)
proof
  let p be Sophie_Germain Prime;
  assume that
A1: p > 2 and
A2: p mod 4 = 3;
  set q = 2*p+1;
A3: q is Safe Prime by Def1,Def2;
  q > 5 by A1,Lm1;
  then
A4: q > 5-3 by XREAL_1:51;
  then 2,q are_coprime by A3,INT_2:28,30;
  then
A5: 2 gcd q = 1 by INT_2:def 3;
  p = (p div 4)*4+3 by A2,INT_1:59;
  then q = (p div 4)*8+7;
  then q mod 8 = 7 mod 8 by NAT_D:21
    .= 7 by NAT_D:24;
  then 2 is_quadratic_residue_mod q by A3,A4,INT_5:43;
  then (2|^((2*p+1-'1) div 2)-1) mod q = 0 by A3,A4,A5,INT_5:20;
  then (2|^((2*p) div 2)-1) mod q = 0 by NAT_D:34;
  then (2|^p-1) mod q = 0 by NAT_D:18;
  then q divides 2|^p-1 by INT_1:62;
  hence thesis by A3;
end;
