reserve k,m,n,p for Nat;

theorem Th1:
  for m, n being Nat holds m gcd n = m gcd (n + m)
proof
  let m, n;
  set a = m gcd n;
  set b = m gcd (n + m);
A1: a divides m by NAT_D:def 5;
A2: b divides m by NAT_D:def 5;
  b divides n + m by NAT_D:def 5;
  then b divides n by A2,NAT_D:10;
  then
A3: b divides a by A2,NAT_D:def 5;
  a divides n by NAT_D:def 5;
  then a divides n + m by A1,NAT_D:8;
  then a divides b by A1,NAT_D:def 5;
  hence thesis by A3,NAT_D:5;
end;
