
theorem Th51:
  for L being domRing, x being Element of L holds BRoots <%-x, 1.L
  %> = ({x}, 1)-bag
proof
  let L be domRing, x be Element of L;
  set r = <%-x, 1.L%>;
  Roots r = {x} by Th45;
  then
A1: support BRoots r = {x} by Def8;
A2: x in {x} 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 = x;
      thus (BRoots r).i = multiplicity(r,i1) by Def8
        .= 1 by A3,Th50
        .= (({x}, 1)-bag).i by A2,A3,Th4;
    end;
    suppose
      i <> x;
      then
A4:   not i in {x} by TARSKI:def 1;
      hence (BRoots r).i = 0 by A1,PRE_POLY:def 7
        .= (({x}, 1)-bag).i by A4,Th3;
    end;
  end;
  hence thesis by PBOOLE:3;
end;
