let X be set ; :: thesis: for L being non empty ZeroStr
for a, b being Element of L holds
( a | (X,L) = b | (X,L) iff a = b )
let L be non empty ZeroStr ; :: thesis: for a, b being Element of L holds
( a | (X,L) = b | (X,L) iff a = b )
let a, b be Element of L; :: thesis: ( a | (X,L) = b | (X,L) iff a = b )
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 ;
set k = (0_ (X,L)) +* ((EmptyBag X),b);
reconsider k = (0_ (X,L)) +* ((EmptyBag X),b) as Function of (Bags X), the carrier of L ;
reconsider k = k as Function of (Bags X),L ;
reconsider k = k as Series of X,L ;
dom ((EmptyBag X) .--> a) = {(EmptyBag X)} by FUNCOP_1:13;
then A1: EmptyBag X in dom ((EmptyBag X) .--> a) by TARSKI:def 1;
dom ((EmptyBag X) .--> b) = {(EmptyBag X)} by FUNCOP_1:13;
then A2: EmptyBag X in dom ((EmptyBag X) .--> b) by TARSKI:def 1;
dom (0_ (X,L)) = dom ((Bags X) --> (0. L)) by POLYNOM1:def 7
.= Bags X by FUNCOP_1:13 ;
then A3: k . (EmptyBag X) = ((0_ (X,L)) +* ((EmptyBag X) .--> b)) . (EmptyBag X) by FUNCT_7:def 3
.= ((EmptyBag X) .--> b) . (EmptyBag X) by A2, FUNCT_4:13
.= b by FUNCOP_1:72 ;
dom (0_ (X,L)) = dom ((Bags X) --> (0. L)) by POLYNOM1:def 7
.= Bags X by FUNCOP_1:13 ;
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_4:13
.= a by FUNCOP_1:72 ;
hence ( a | (X,L) = b | (X,L) iff a = b ) by A3; :: thesis: verum