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 a1,a2 be Element of L, b be bag of X holds
      Monom(a1,b) + Monom(a2,b) = Monom(a1+a2,b)
proof
  let L be add-associative right_zeroed right_complementable non
    empty doubleLoopStr;
  let a1,a2 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(a1,b) + Monom(a2,b)).x = Monom(a1+a2,b).x
  proof
    assume
A2: x in Bags X;
    then reconsider s=x as bag of X;
    per cases;
    suppose
A3:   x=b;
      thus (Monom(a1,b) + Monom(a2,b)).x =
      (Monom(a1,b).s) + (Monom(a2,b).s) by POLYNOM1:15
      .=(Monom(a1,b).s) + (a2) by A1,A2,A3,FUNCT_7:31
      .= a1 + a2 by A1,A2,A3,FUNCT_7:31
      .= Monom(a1+a2,b).x by A1,A2,A3,FUNCT_7:31;
    end;
    suppose
A4:   x<>b;
      thus ( Monom(a1,b) + Monom(a2,b)).x =
      (Monom(a1,b).s) + (Monom(a2,b).s) by POLYNOM1:15
      .=(Monom(a1,b).s) + (0_(X,L).s) by A4,FUNCT_7:32
      .=(0_(X,L).s) + (0_(X,L).s) by A4,FUNCT_7:32
      .=(0_(X,L).s) + (0.L) by POLYNOM1:22
      .=(Monom(a1+a2,b).x) by A4,FUNCT_7:32;
    end;
  end;
  hence thesis;
end;
