let X be set ; :: thesis: for L being non empty ZeroStr
for a being Element of L holds
( term (a | X,L) = EmptyBag X & coefficient (a | X,L) = a )
let L be non empty ZeroStr ; :: thesis: for a being Element of L holds
( term (a | X,L) = EmptyBag X & coefficient (a | X,L) = a )
let a be Element of L; :: thesis: ( term (a | X,L) = EmptyBag X & coefficient (a | X,L) = a )
set m = (0_ X,L) +* (EmptyBag X),a;
reconsider m = (0_ X,L) +* (EmptyBag X),a as Function of (Bags X),the carrier of L ;
reconsider m = m as Function of (Bags X),L ;
reconsider m = m as Series of X,L ;
dom ((EmptyBag X) .--> a) = {(EmptyBag X)} by FUNCOP_1:19;
then A1: EmptyBag X in dom ((EmptyBag X) .--> a) by TARSKI:def 1;
dom (0_ X,L) = dom ((Bags X) --> (0. L)) by POLYNOM1:def 24
.= Bags X by FUNCOP_1:19 ;
then m . (EmptyBag X) = ((0_ X,L) +* ((EmptyBag X) .--> a)) . (EmptyBag X) by FUNCT_7:def 3
.= ((EmptyBag X) .--> a) . (EmptyBag X) by A1, FUNCT_4:14
.= a by FUNCOP_1:87 ;
hence ( term (a | X,L) = EmptyBag X & coefficient (a | X,L) = a ) by Lm2; :: thesis: verum