begin
Lm1:
for i, I, X being set
for M being ManySortedSet of I st i in I holds
dom (M +* (i .--> X)) = I
Lm2:
for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A \/ B)) . i = bool ((A . i) \/ (B . i))
Lm3:
for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A /\ B)) . i = bool ((A . i) /\ (B . i))
Lm4:
for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A \ B)) . i = bool ((A . i) \ (B . i))
Lm5:
for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A \+\ B)) . i = bool ((A . i) \+\ (B . i))
begin
Lm6:
for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(union (A \/ B)) . i = union ((A . i) \/ (B . i))
Lm7:
for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(union (A /\ B)) . i = union ((A . i) /\ (B . i))