reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem Th41:
  3 is prime
proof
  for n being Nat holds n divides 3 implies n = 1 or n = 3
  proof
    let n be Nat;
    assume
A1: n divides 3;
A2: n <> 2
    proof
A3:   3 mod 2 = (2*1 + 1) mod 2 .= 1 mod 2 by NAT_D:21
        .= 1 by NAT_D:24;
      assume n = 2;
      hence contradiction by A1,A3,Th6;
    end;
    n <= 3 by A1,NAT_D:7;
    then n < 2 + 1 or n = 3 by XXREAL_0:1;
    then n <= 2 or n = 3 by NAT_1:13;
    then n < 1 + 1 or n = 2 or n = 3 by XXREAL_0:1;
    then n <= 1 or n = 2 or n = 3 by NAT_1:13;
    then
A4: n < 1 or n = 1 or n = 2 or n = 3 by XXREAL_0:1;
    n <> 0 by A1;
    hence thesis by A4,A2,NAT_1:14;
  end;
  hence thesis by INT_2:def 4;
end;
