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
  m gcd (n gcd k) = (m gcd n) gcd k
proof
  set M = n gcd k;
  set K = m gcd M;
  set N = m gcd n;
  set L = N gcd k;
A1: K divides M by NAT_D:def 5;
A2: K divides m by NAT_D:def 5;
  M divides n by NAT_D:def 5;
  then K divides n by A1,NAT_D:4;
  then
A3: K divides N by A2,NAT_D:def 5;
A4: L divides N by NAT_D:def 5;
A5: L divides k by NAT_D:def 5;
  N divides n by NAT_D:def 5;
  then L divides n by A4,NAT_D:4;
  then
A6: L divides M by A5,NAT_D:def 5;
  N divides m by NAT_D:def 5;
  then L divides m by A4,NAT_D:4;
  then
A7: L divides K by A6,NAT_D:def 5;
  M divides k by NAT_D:def 5;
  then K divides k by A1,NAT_D:4;
  then K divides L by A3,NAT_D:def 5;
  hence thesis by A7,NAT_D:5;
end;
