let K be Field; :: thesis: for R1, R2 being FinSequence of K st R1,R2 are_fiberwise_equipotent holds
the multF of K $$ R1 = the multF of K $$ R2
defpred S1[ Nat] means for f, g being FinSequence of K st f,g are_fiberwise_equipotent & len f = $1 holds
the multF of K $$ f = the multF of K $$ g;
A1:
S1[ 0 ]
A2:
for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be
Nat;
:: thesis: ( S1[n] implies S1[n + 1] )
assume A3:
S1[
n]
;
:: thesis: S1[n + 1]
let f,
g be
FinSequence of
K;
:: thesis: ( f,g are_fiberwise_equipotent & len f = n + 1 implies the multF of K $$ f = the multF of K $$ g )
assume A4:
(
f,
g are_fiberwise_equipotent &
len f = n + 1 )
;
:: thesis: the multF of K $$ f = the multF of K $$ g
then A5:
(
rng f = rng g &
len f = len g )
by CLASSES1:83, RFINSEQ:16;
0 + 1
<= n + 1
by NAT_1:13;
then A6:
n + 1
in dom f
by A4, FINSEQ_3:27;
then A7:
f . (n + 1) in rng f
by FUNCT_1:def 5;
rng f c= the
carrier of
K
by FINSEQ_1:def 4;
then reconsider a =
f . (n + 1) as
Element of
K by A7;
a in rng g
by A5, A6, FUNCT_1:def 5;
then consider m being
Nat such that A8:
(
m in dom g &
g . m = a )
by FINSEQ_2:11;
A9:
g = (g | m) ^ (g /^ m)
by RFINSEQ:21;
A10:
( 1
<= m &
m <= len g )
by A8, FINSEQ_3:27;
then
max 0 ,
(m - 1) = m - 1
by FINSEQ_2:4;
then reconsider m1 =
m - 1 as
Element of
NAT by FINSEQ_2:5;
set gg =
g /^ m;
set gm =
g | m;
A11:
len (g | m) = m
by A10, FINSEQ_1:80;
A12:
m = m1 + 1
;
m in NAT
by ORDINAL1:def 13;
then
m in Seg m
by A10;
then
(
(g | m) . m = a &
m in dom g )
by A8, RFINSEQ:19;
then A13:
g | m = ((g | m) | m1) ^ <*a*>
by A11, A12, RFINSEQ:20;
m1 <= m
by A12, NAT_1:11;
then A14:
Seg m1 c= Seg m
by FINSEQ_1:7;
A15:
(g | m) | m1 =
g | ((Seg m) /\ (Seg m1))
by RELAT_1:100
.=
g | m1
by A14, XBOOLE_1:28
;
reconsider n1 =
n as
Element of
NAT by ORDINAL1:def 13;
set fn =
f | n1;
A16:
len (f | n1) = n
by A4, FINSEQ_1:80, NAT_1:11;
A17:
f = (f | n1) ^ <*a*>
by A4, RFINSEQ:20;
now let x be
set ;
:: thesis: card (Coim (f | n1),x) = card (Coim ((g | m1) ^ (g /^ m)),x)
card (Coim f,x) = card (Coim g,x)
by A4, CLASSES1:def 9;
then (card ((f | n1) " {x})) + (card (<*a*> " {x})) =
card ((((g | m1) ^ <*a*>) ^ (g /^ m)) " {x})
by A9, A13, A15, A17, FINSEQ_3:63
.=
(card (((g | m1) ^ <*a*>) " {x})) + (card ((g /^ m) " {x}))
by FINSEQ_3:63
.=
((card ((g | m1) " {x})) + (card (<*a*> " {x}))) + (card ((g /^ m) " {x}))
by FINSEQ_3:63
.=
((card ((g | m1) " {x})) + (card ((g /^ m) " {x}))) + (card (<*a*> " {x}))
.=
(card (((g | m1) ^ (g /^ m)) " {x})) + (card (<*a*> " {x}))
by FINSEQ_3:63
;
hence
card (Coim (f | n1),x) = card (Coim ((g | m1) ^ (g /^ m)),x)
;
:: thesis: verum end;
then
f | n1,
(g | m1) ^ (g /^ m) are_fiberwise_equipotent
by CLASSES1:def 9;
then
the
multF of
K $$ (f | n1) = the
multF of
K $$ ((g | m1) ^ (g /^ m))
by A3, A16;
hence the
multF of
K $$ f =
(the multF of K $$ ((g | m1) ^ (g /^ m))) * (the multF of K $$ <*a*>)
by A17, Th30
.=
((the multF of K $$ (g | m1)) * (the multF of K $$ (g /^ m))) * (the multF of K $$ <*a*>)
by Th30
.=
((the multF of K $$ (g | m1)) * (the multF of K $$ <*a*>)) * (the multF of K $$ (g /^ m))
by GROUP_1:def 4
.=
(the multF of K $$ (g | m)) * (the multF of K $$ (g /^ m))
by A13, A15, Th30
.=
the
multF of
K $$ g
by A9, Th30
;
:: thesis: verum
end;
A18:
for n being Nat holds S1[n]
from NAT_1:sch 2(A1, A2);
let R1, R2 be FinSequence of K; :: thesis: ( R1,R2 are_fiberwise_equipotent implies the multF of K $$ R1 = the multF of K $$ R2 )
assume A19:
R1,R2 are_fiberwise_equipotent
; :: thesis: the multF of K $$ R1 = the multF of K $$ R2
len R1 = len R1
;
hence
the multF of K $$ R1 = the multF of K $$ R2
by A18, A19; :: thesis: verum