let X be set ; :: thesis: for L being non empty ZeroStr holds (0. L) | (X,L) = 0_ (X,L)
let L be non empty ZeroStr ; :: thesis: (0. L) | (X,L) = 0_ (X,L)
set o1 = (0. L) | (X,L);
set o2 = 0_ (X,L);

now :: thesis: for x being object st x in Bags X holds
((0. L) | (X,L)) . x = (0_ (X,L)) . x

set m = (0_ (X,L)) +* ((EmptyBag X),(0. L));
let x be object ; :: thesis: ( x in Bags X implies ((0. L) | (X,L)) . b₁ = (0_ (X,L)) . b₁ )
reconsider m = (0_ (X,L)) +* ((EmptyBag X),(0. L)) 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 ;
assume x in Bags X ; :: thesis: ((0. L) | (X,L)) . b₁ = (0_ (X,L)) . b₁
then reconsider x9 = x as bag of X ;
A1: dom (0_ (X,L)) = dom ((Bags X) --> (0. L)) by POLYNOM1:def 8
.= Bags X ;
then A2: m = (0_ (X,L)) +* ((EmptyBag X) .--> (0. L)) by FUNCT_7:def 3;
A3: EmptyBag X in dom ((EmptyBag X) .--> (0. L)) by TARSKI:def 1;
A4: m . (EmptyBag X) = ((0_ (X,L)) +* ((EmptyBag X) .--> (0. L))) . (EmptyBag X) by A1, FUNCT_7:def 3
.= ((EmptyBag X) .--> (0. L)) . (EmptyBag X) by A3, FUNCT_4:13
.= 0. L by FUNCOP_1:72 ;

per cases ( x9 = EmptyBag X or x9 <> EmptyBag X ) ;

suppose x9 = EmptyBag X ; :: thesis: ((0. L) | (X,L)) . b₁ = (0_ (X,L)) . b₁

hence ((0. L) | (X,L)) . x = (0_ (X,L)) . x by A4, POLYNOM1:22; :: thesis: verum

end;

suppose x9 <> EmptyBag X ; :: thesis: ((0. L) | (X,L)) . b₁ = (0_ (X,L)) . b₁

then not x9 in dom ((EmptyBag X) .--> (0. L)) by TARSKI:def 1;
hence ((0. L) | (X,L)) . x = (0_ (X,L)) . x by A2, FUNCT_4:11; :: thesis: verum

end;

end;

end;

hence (0. L) | (X,L) = 0_ (X,L) ; :: thesis: verum