
theorem
  for L being Field
  for a being Element of L
  for b being non zero Element of L holds
  BRoots <%a,b%> = ({-a/b},1)-bag
  proof
    let L be Field;
    let a be Element of L;
    let b be non zero Element of L;
    set r = <%a,b%>;
    Roots r = {-a/b} by Th10;
    then
A1: support BRoots r = {-a/b} by UPROOTS:def 9;
A2: -a/b in {-a/b} by TARSKI:def 1;
    now
      let i be object;
      assume i in the carrier of L;
      then reconsider i1 = i as Element of L;
      per cases;
      suppose
A3:     i = -a/b;
        thus (BRoots r).i = multiplicity(r,i1) by UPROOTS:def 9
        .= 1 by A3,Th11
        .= (({-a/b},1)-bag).i by A2,A3,UPROOTS:7;
      end;
      suppose i <> -a/b;
        then
A4:     not i in {-a/b} by TARSKI:def 1;
        hence (BRoots r).i = 0 by A1,PRE_POLY:def 7
        .= (({-a/b},1)-bag).i by A4,UPROOTS:6;
      end;
    end;
    hence thesis by PBOOLE:3;
  end;
