
theorem
  for p be Prime st p <> 2 holds Mersenne(p) mod 8 = 7
proof
  let p be Prime;
A1: p > 1 by INT_2:def 4;
  then
A2: p >= 1+1 by NAT_1:13;
  assume p <> 2;
  then p > 2 + 0 by A2,XXREAL_0:1;
  then p-2 > 2-2 by XREAL_1:14;
  then
A3: p-'2 > 0 by A2,XREAL_1:233;
A4: p-'1 > 0 by A1,NAT_D:49;
  Mersenne(p) mod 8 = 2*2|^(p-'1)-1 mod 8 by PEPIN:26
    .= ((2*2|^(p-'1) mod 2*4)-(1 mod 8)) mod 8 by INT_6:7
    .= (2*(2|^(p-'1) mod 4)-(1 mod 8)) mod 8 by INT_4:20
    .= (2*(2*2|^(p-'1-'1) mod 2*2)-(1 mod 8)) mod 8 by A4,PEPIN:26
    .= (2*(2*2|^(p-'2) mod 2*2)-(1 mod 8)) mod 8 by NAT_D:45
    .= (2*(2*( 2|^(p-'2) mod 2))-(1 mod 8)) mod 8 by INT_4:20
    .= (4*(2|^(p-'2) mod 2)-1) mod 8 by PEPIN:5
    .= (4*0-1) mod 8 by A3,PEPIN:36
    .= (-1+8*1) mod 8 by NAT_D:61
    .= 7 by NAT_D:24;
  hence thesis;
end;
