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 Th13:
  for a,b being Nat, m being non zero Nat holds
  a,b are_coprime implies
  Product Sgm {p where p is Prime: p divides m & p divides a},
  Product Sgm {q where q is Prime: q divides m & q divides b} are_coprime
  proof
    let a,b;
    let m be non zero Nat such that
A1: a,b are_coprime;
    set A = {p where p is Prime: p divides m & p divides a};
    set B = {q where q is Prime: q divides m & q divides b};
    assume not thesis;
    then consider P being Prime such that
A2: P divides Product Sgm A and
A3: P divides Product Sgm B by PYTHTRIP:def 2;
    A is included_in_Seg by Lm21;
    then P in A by A2,Th12,Lm23;
    then
A4: ex p being Prime st P = p & p divides m & p divides a;
    B is included_in_Seg by Lm21;
    then P in B by A3,Th12,Lm23;
    then ex q being Prime st P = q & q divides m & q divides b;
    hence contradiction by A1,A4,PYTHTRIP:def 2;
  end;
