reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  for a be non trivial Nat holds
  ex n be prime Nat st n divides a
  proof
    let a be non trivial Nat;
    per cases;
    suppose a is prime;
      hence thesis;
    end;
    suppose not a is prime; then
      ex n be Element of NAT st n > 1 & n*n <= a & n is prime &
      n divides a by NAT_4:14,Def0;
      hence thesis;
    end;
  end;
