reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem
  for L be add-associative right_zeroed right_complementable
           non empty doubleLoopStr
    for a be Element of L, b be bag of X holds
      Monom(-a,b) = -Monom(a,b)
proof
  let L be add-associative right_zeroed right_complementable non
    empty doubleLoopStr;
  let a be Element of L, b be bag of X;
A1: dom 0_(X,L) = Bags X by FUNCT_2:def 1;
  x in Bags X implies Monom(-a,b).x = (-Monom(a,b)).x
  proof
    assume
A2: x in Bags X;
    then reconsider s=x as bag of X;
    per cases;
    suppose
A3:   x=b;
      hence Monom(-a,b).x = -a by A1,A2,FUNCT_7:31
      .= -(Monom(a,b).s) by A3,A1,A2,FUNCT_7:31
      .= (-Monom(a,b)).x by POLYNOM1:17;
    end;
    suppose
A4:   x<>b;
      hence Monom(-a,b).x = 0_(X,L).x by FUNCT_7:32
      .= -(0.L) by A2,POLYNOM1:22
      .= -(0_(X,L).s) by POLYNOM1:22
      .= -(Monom(a,b).s) by A4,FUNCT_7:32
      .= (-Monom(a,b)).x by POLYNOM1:17;
    end;
  end;
  hence thesis;
end;
