let n be set ; :: thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Series of n,L holds p - (0_ n,L) = p
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for p being Series of n,L holds p - (0_ n,L) = p
let p be Series of n,L; :: thesis: p - (0_ n,L) = p
reconsider pp = p - (0_ n,L) as Function of (Bags n),the carrier of L ;

now

let b be Element of Bags n; :: thesis: pp . b = p . b
thus pp . b = (p + (- (0_ n,L))) . b by POLYNOM1:def 23
.= (p . b) + ((- (0_ n,L)) . b) by POLYNOM1:def 21
.= (p . b) + (- ((0_ n,L) . b)) by POLYNOM1:def 22
.= (p . b) + (- (0. L)) by POLYNOM1:81
.= (p . b) - (0. L) by RLVECT_1:def 12
.= p . b by RLVECT_1:26 ; :: thesis: verum

end;

hence p - (0_ n,L) = p by FUNCT_2:113; :: thesis: verum