let R1, R2 be real-valued FinSequence; ( R1,R2 are_fiberwise_equipotent implies Product R1 = Product R2 )
defpred S1[ Nat] means for f, g being FinSequence of REAL st f,g are_fiberwise_equipotent & len f = $1 holds
Product f = Product g;
assume A1:
R1,R2 are_fiberwise_equipotent
; Product R1 = Product R2
A2:
len R1 = len R1
;
A3:
for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be
Nat;
( S1[n] implies S1[n + 1] )
assume A4:
S1[
n]
;
S1[n + 1]
let f,
g be
FinSequence of
REAL ;
( f,g are_fiberwise_equipotent & len f = n + 1 implies Product f = Product g )
assume that A5:
f,
g are_fiberwise_equipotent
and A6:
len f = n + 1
;
Product f = Product g
set a =
f . (n + 1);
A7:
rng f = rng g
by A5, CLASSES1:75;
0 + 1
<= n + 1
by NAT_1:13;
then
n + 1
in dom f
by A6, FINSEQ_3:25;
then
f . (n + 1) in rng g
by A7, FUNCT_1:def 3;
then consider m being
Nat such that A8:
m in dom g
and A9:
g . m = f . (n + 1)
by FINSEQ_2:10;
set gg =
g /^ m;
set gm =
g | m;
A11:
1
<= m
by A8, FINSEQ_3:25;
max (
0,
(m - 1))
= m - 1
by FINSEQ_2:4, A8, FINSEQ_3:25;
then reconsider m1 =
m - 1 as
Element of
NAT by FINSEQ_2:5;
m = m1 + 1
;
then A13:
Seg m1 c= Seg m
by FINSEQ_1:5, NAT_1:11;
m in Seg m
by A11;
then WW:
f . (n + 1) = (g | m) . (m1 + 1)
by A8, A9, RFINSEQ:6;
m <= len g
by A8, FINSEQ_3:25;
then
len (g | m) = m
by FINSEQ_1:59;
then A14:
g | m = ((g | m) | m1) ^ <*(f . (n + 1))*>
by WW, RFINSEQ:7;
set fn =
f | n;
A15:
g = (g | m) ^ (g /^ m)
by RFINSEQ:8;
A16:
(g | m) | m1 =
g | ((Seg m) /\ (Seg m1))
by RELAT_1:71
.=
g | m1
by A13, XBOOLE_1:28
;
A17:
f = (f | n) ^ <*(f . (n + 1))*>
by A6, RFINSEQ:7;
now for x being object holds card (Coim ((f | n),x)) = card (Coim (((g | m1) ^ (g /^ m)),x))let x be
object ;
card (Coim ((f | n),x)) = card (Coim (((g | m1) ^ (g /^ m)),x))
card (Coim (f,x)) = card (Coim (g,x))
by A5;
then (card ((f | n) " {x})) + (card (<*(f . (n + 1))*> " {x})) =
card ((((g | m1) ^ <*(f . (n + 1))*>) ^ (g /^ m)) " {x})
by A15, A14, A16, A17, FINSEQ_3:57
.=
(card (((g | m1) ^ <*(f . (n + 1))*>) " {x})) + (card ((g /^ m) " {x}))
by FINSEQ_3:57
.=
((card ((g | m1) " {x})) + (card (<*(f . (n + 1))*> " {x}))) + (card ((g /^ m) " {x}))
by FINSEQ_3:57
.=
((card ((g | m1) " {x})) + (card ((g /^ m) " {x}))) + (card (<*(f . (n + 1))*> " {x}))
.=
(card (((g | m1) ^ (g /^ m)) " {x})) + (card (<*(f . (n + 1))*> " {x}))
by FINSEQ_3:57
;
hence
card (Coim ((f | n),x)) = card (Coim (((g | m1) ^ (g /^ m)),x))
;
verum end;
then A18:
f | n,
(g | m1) ^ (g /^ m) are_fiberwise_equipotent
;
len (f | n) = n
by A6, FINSEQ_1:59, NAT_1:11;
then
Product (f | n) = Product ((g | m1) ^ (g /^ m))
by A4, A18;
hence Product f =
(Product ((g | m1) ^ (g /^ m))) * (Product <*(f . (n + 1))*>)
by A17, RVSUM_1:97
.=
((Product (g | m1)) * (Product (g /^ m))) * (Product <*(f . (n + 1))*>)
by RVSUM_1:97
.=
((Product (g | m1)) * (Product <*(f . (n + 1))*>)) * (Product (g /^ m))
.=
(Product (g | m)) * (Product (g /^ m))
by A14, A16, RVSUM_1:97
.=
Product g
by A15, RVSUM_1:97
;
verum
end;
A19:
S1[ 0 ]
A4:
for n being Nat holds S1[n]
from NAT_1:sch 2(A19, A3);
( R1 is FinSequence of REAL & R2 is FinSequence of REAL )
by RVSUM_1:145;
hence
Product R1 = Product R2
by A1, A2, A4; verum