reserve a,b,c,k,l,m,n for Nat,
  i,j,x,y for Integer;

theorem Th2:
  for k,n be Nat holds k <> 0 & k < n & n is prime implies k,n are_coprime
proof
  let k,n be Nat;
  assume that
A1: k <> 0 & k < n and
A2: n is prime;
A3: (k gcd n) divides n by NAT_D:def 5;
  per cases by A2,A3,INT_2:def 4;
  suppose
    k gcd n = 1;
    hence thesis by INT_2:def 3;
  end;
  suppose
    k gcd n = n;
    then n divides k by NAT_D:def 5;
    hence thesis by A1,NAT_D:7;
  end;
end;
