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
  k>0 & a+b > k & a+b divides k*a implies not a,b are_coprime
  proof
    assume
    A1: k>0 & a+b > k & a+b divides k*a;
    per cases;
    suppose
      B1: b > 0;
      B2: not a+b, a are_coprime by A1,INT_2:25,27;
      assume not thesis; then
      1 = b+1*a gcd a by B1,EULER_1:16;
      hence contradiction by B2;
    end;
    suppose
      b = 0;
      hence thesis by A1,NAT_1:14;
    end;
  end;
