
theorem
  for p be Prime st p <> 2 holds Mersenne(p) mod 2*p = 1
proof
  let p be Prime;
  assume
A1: p <> 2;
  p > 1 by INT_2:def 4;
  then 2*p > 2*1 by XREAL_1:68;
  then
A2: 2*p > 2-1 by XREAL_1:51;
  Mersenne(p) mod 2*p = 2*2|^(p-'1)-1 mod 2*p by PEPIN:26
    .= ((2*2|^(p-'1) mod 2*p)-(1 mod 2*p)) mod 2*p by INT_6:7
    .= (2*(2|^(p-'1) mod p)-(1 mod 2*p)) mod 2*p by INT_4:20
    .= (2*1-(1 mod 2*p)) mod 2*p by A1,INT_2:28,30,PEPIN:37
    .= (2*1-1) mod 2*p by A2,PEPIN:5;
  hence thesis by A2,PEPIN:5;
end;
