reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem
  (n gcd m) lcm (n gcd k) divides n gcd (m lcm k)
proof
  set M = m lcm k;
  set N = n gcd k;
  set P = n gcd m;
  set L = P lcm N;
A1: N divides k by NAT_D:def 5;
  k divides M by NAT_D:def 4;
  then
A2: N divides M by A1,NAT_D:4;
A3: P divides m by NAT_D:def 5;
  m divides M by NAT_D:def 4;
  then P divides M by A3,NAT_D:4;
  then
A4: L divides M by A2,NAT_D:def 4;
A5: P divides n by NAT_D:def 5;
  N divides n by NAT_D:def 5;
  then L divides n by A5,NAT_D:def 4;
  hence thesis by A4,NAT_D:def 5;
end;
