reserve a,b,c,k,k9,m,n,n9,p,p9 for Nat;
reserve i,i9 for Integer;

theorem Th8:
  (k*m) gcd (k*n) = k*(m gcd n)
proof
  per cases;
  suppose
A1: k <> 0;
    k divides k*m & k divides k*n;
    then k divides (k*m) gcd (k*n) by NAT_D:def 5;
    then consider k9 be Nat such that
A2: (k*m) gcd (k*n) = k*k9 by NAT_D:def 3;
    reconsider k9 as Element of NAT by ORDINAL1:def 12;
    now
      k*k9 divides k*m by A2,NAT_D:def 5;
      hence k9 divides m by A1,Th7;
      k*k9 divides k*n by A2,NAT_D:def 5;
      hence k9 divides n by A1,Th7;
      let p be Nat;
      reconsider p9=p as Element of NAT by ORDINAL1:def 12;
      assume p divides m & p divides n;
      then k*p9 divides k*m & k*p9 divides k*n by Th7;
      then k*p divides k*k9 by A2,NAT_D:def 5;
      then p9 divides k9 by A1,Th7;
      hence p divides k9;
    end;
    hence thesis by A2,NAT_D:def 5;
  end;
  suppose
    k = 0;
    hence thesis by NEWTON:52;
  end;
end;
