reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem
  a+b divides c & a,c are_coprime implies a,b are_coprime
  proof
    assume
    A1: a+b divides c & a,c are_coprime; then
    consider k such that
    A2: c = (a+b)*k by NAT_D:def 3;
    1 = a gcd ((b*k) + k*a) by A1,A2
    .= a gcd b*k by Th5; then
    a*1, k*b are_coprime;
    hence thesis by NEWTON01:41;
  end;
