reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th19:
  p is prime implies p = 2 or p mod 4 = 1 or p mod 4 = 3
  proof
    assume that
A1: p is prime and
A2: p <> 2 & p mod 4 <> 1 & p mod 4 <> 3;
    set D = p div 4;
    p mod (3+1) = 0 or ... or p mod (3+1) = 3 by NUMBER03:11;
    then per cases by A2;
    suppose p mod 4 = 0;
      then 4 divides p by PEPIN:6;
      hence thesis by A1,INT_2:29;
    end;
    suppose p mod 4 = 2;
      then p = 4*D+2 by INT_1:59;
      then p = 2*(2*D+1);
      then 2 divides p;
      hence thesis by A2,A1;
    end;
  end;
