let n be set ; :: thesis: for L being non empty left_zeroed addLoopStr
for p being Series of n,L holds (0_ (n,L)) + p = p
let L be non empty left_zeroed addLoopStr ; :: thesis: for p being Series of n,L holds (0_ (n,L)) + p = p
let p be Series of n,L; :: thesis: (0_ (n,L)) + p = p
reconsider ls = (0_ (n,L)) + p, p9 = p as Function of (Bags n), the carrier of L ;

now :: thesis: for b being Element of Bags n holds ls . b = p9 . b

let b be Element of Bags n; :: thesis: ls . b = p9 . b
thus ls . b = ((0_ (n,L)) . b) + (p . b) by POLYNOM1:15
.= (0. L) + (p9 . b) by POLYNOM1:22
.= p9 . b by ALGSTR_1:def 2 ; :: thesis: verum

end;

hence (0_ (n,L)) + p = p by FUNCT_2:63; :: thesis: verum