reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;
reserve z for Complex;

theorem Th15:
  for a,b being Nat, m being non zero Nat holds
  a,b are_coprime implies
  Product Sgm {q where q is Prime: q divides m & q divides b},
  Product Sgm
     {r where r is Prime: r divides m & not r divides a & not r divides b}
  are_coprime
  proof
    let a,b;
    let m be non zero Nat;
    set A =
    {r where r is Prime: r divides m & not r divides a & not r divides b};
    set B =
    {r where r is Prime: r divides m & not r divides b & not r divides a};
    A = B
    proof
      hereby
        let x be object;
        assume x in A;
        then ex r being Prime st x = r &
        r divides m & not r divides a & not r divides b;
        hence x in B;
      end;
      let x be object;
      assume x in B;
      then ex r being Prime st x = r &
      r divides m & not r divides b & not r divides a;
      hence x in A;
    end;
    hence thesis by Th14;
  end;
