
theorem Th10:
  for L being Field
  for a being Element of L
  for b being non zero Element of L holds
  Roots <%a,b%> = {-a/b}
  proof
    let L be Field;
    let a be Element of L;
    let b be non zero Element of L;
    set p = <%a,b%>;
    thus Roots p c= {-a/b}
    proof
      let x be object;
      assume
A1:   x in Roots p;
      then reconsider x as Element of L;
      x is_a_root_of p by A1,POLYNOM5:def 10;
      then x = -a/b by Th9;
      hence thesis by TARSKI:def 1;
    end;
    let x be object;
    assume x in {-a/b};
    then
A2: x = -a/b by TARSKI:def 1;
    -a/b is_a_root_of p by Th9;
    hence thesis by A2,POLYNOM5:def 10;
  end;
