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

theorem Th6:
  for i,n be Nat st i <> 0 holds i divides n iff n mod i = 0
proof
  let i,n be Nat;
  assume
A1: i <> 0;
A2: n mod i = 0 implies i divides n
  proof
    assume n mod i = 0;
    then ex t being Nat st n = i*t + 0 & 0 < i by A1,NAT_D:def 2;
    hence thesis;
  end;
  i divides n implies n mod i = 0
  proof
    assume i divides n;
    then ex t being Nat st n = i*t by NAT_D:def 3;
    hence thesis by NAT_D:13;
  end;
  hence thesis by A2;
end;
