reserve x,y,z for Real,
  a,b,c,d,e,f,g,h for Nat,
  k,l,m,n,m1,n1,m2,n2 for Integer,
  q for Rational;

theorem Th10:
  k,l are_coprime implies k|^a,l are_coprime
proof
  defpred P[Nat] means k|^$1,l are_coprime;
  assume
A1: k,l are_coprime;
A2: for a st P[a] holds P[a+1]
  proof
    let a;
    assume k|^a,l are_coprime;
    then k*(k|^a),l are_coprime by A1,INT_2:26;
    hence thesis by NEWTON:6;
  end;
  k|^0=1 by NEWTON:4;
  then
A3: P[0] by Th9;
  for a holds P[a] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
