
theorem
  for p be Safe Prime st p <> 7 holds p mod 6 = 5
proof
  let p be Safe Prime;
  assume
A1: p <> 7;
A2: 4*p mod 6 = 2*(2*p) mod 2*3 .= 2*((2*p) mod 3) by INT_4:20
    .= 2*(((2 mod 3)*(p mod 3)) mod 3) by NAT_D:67
    .= 2*((2*(p mod 3)) mod 3) by NAT_D:24
    .= 2*((2*2) mod 3) by A1,Th4
    .= 2*((1+3*1) mod 3)
    .= 2*(1 mod 3) by NAT_D:61
    .= 2*1 by PEPIN:5;
A3: 3*p mod 6 = 3*p mod 3*2 .= 3*(p mod 2) by INT_4:20
    .= 3*1 by Th3;
  p mod 6 = (p+6*p) mod 6 by NAT_D:61
    .= (3*p+4*p) mod 6
    .= (3*1+2*1) mod 6 by A3,A2,NAT_D:66
    .= 5 by NAT_D:24;
  hence thesis;
end;
