let n be set ; :: thesis: for L being non trivial ZeroStr
for a being non zero Element of L
for b being bag of n holds term (Monom (a,b)) = b
let L be non trivial ZeroStr ; :: thesis: for a being non zero Element of L
for b being bag of n holds term (Monom (a,b)) = b
let a be non zero Element of L; :: thesis: for b being bag of n holds term (Monom (a,b)) = b
let b be bag of n; :: thesis: term (Monom (a,b)) = b
set m = (0_ (n,L)) +* (b,a);
reconsider m = (0_ (n,L)) +* (b,a) as Function of (Bags n), the carrier of L ;
reconsider m = m as Function of (Bags n),L ;
reconsider m = m as Series of n,L ;
A1: b in Bags n by PRE_POLY:def 12;
dom (b .--> a) = {b} by FUNCOP_1:13;
then A2: b in dom (b .--> a) by TARSKI:def 1;
dom (0_ (n,L)) = dom ((Bags n) --> (0. L)) by POLYNOM1:def 7
.= Bags n by FUNCOP_1:13 ;
then m . b = ((0_ (n,L)) +* (b .--> a)) . b by A1, FUNCT_7:def 3
.= (b .--> a) . b by A2, FUNCT_4:13
.= a by FUNCOP_1:72 ;
then m . b <> 0. L ;
hence term (Monom (a,b)) = b by Def6; :: thesis: verum