
theorem
  for p be Safe Prime st p > 7 holds p mod 12 = 11
proof
  let p be Safe Prime;
  assume
A1: p > 7;
  then p > 7-2 by XREAL_1:51;
  then
A2: p mod 4 = 3 by Th5;
A3: 9*p mod 12 = 3*(3*p) mod 3*4 .= 3*((3*p) mod 4) by INT_4:20
    .= 3*(((3 mod 4)*(p mod 4)) mod 4) by NAT_D:67
    .= 3*((3 * 3) mod 4) by A2,NAT_D:24
    .= 3*((1+4*2) mod 4)
    .= 3*(1 mod 4) by NAT_D:61
    .= 3*1 by PEPIN:5;
A4: 4*p mod 12 = 4*p mod 4*3 .= 4*(p mod 3) by INT_4:20
    .= 4*2 by A1,Th4;
  p mod 12 = (p+12*p) mod 12 by NAT_D:61
    .= (4*p+9*p) mod 12
    .= (4*2+3*1) mod 12 by A4,A3,NAT_D:66
    .= 11 by NAT_D:24;
  hence thesis;
end;
