begin
theorem Th2:
for
x,
y being
R_eal for
a,
b being
Real st
x = a &
y = b holds
x / y = a / b
begin
Lm1:
for F being FinSequence of ExtREAL
for x being Element of ExtREAL holds Sum (F ^ <*x*>) = (Sum F) + x
Lm2:
for F being FinSequence of ExtREAL st not -infty in rng F holds
Sum F <> -infty
Lm3:
for n, q being Nat st q in Seg (n + 1) holds
ex p being Permutation of (Seg (n + 1)) st
for i being Element of NAT st i in Seg (n + 1) holds
( ( i < q implies p . i = i ) & ( i = q implies p . i = n + 1 ) & ( i > q implies p . i = i - 1 ) )
Lm4:
for n, q being Nat
for s, p being Permutation of (Seg (n + 1))
for s9 being FinSequence of Seg (n + 1) st s9 = s & q = s . (n + 1) & ( for i being Element of NAT st i in Seg (n + 1) holds
( ( i < q implies p . i = i ) & ( i = q implies p . i = n + 1 ) & ( i > q implies p . i = i - 1 ) ) ) holds
p * (s9 | n) is Permutation of (Seg n)
Lm5:
for n, q being Nat
for p being Permutation of (Seg (n + 1))
for F, H being FinSequence of ExtREAL st F = H * p & q in Seg (n + 1) & len H = n + 1 & not -infty in rng H & ( for i being Element of NAT st i in Seg (n + 1) holds
( ( i < q implies p . i = i ) & ( i = q implies p . i = n + 1 ) & ( i > q implies p . i = i - 1 ) ) ) holds
Sum F = Sum H