let v1, v2 be Element of V; ( ex F being FinSequence of the carrier of V st
( F is one-to-one & rng F = Carrier L & v1 = Sum (L (#) F) ) & ex F being FinSequence of the carrier of V st
( F is one-to-one & rng F = Carrier L & v2 = Sum (L (#) F) ) implies v1 = v2 )
given F1 being FinSequence of the carrier of V such that A4:
F1 is one-to-one
and
A5:
rng F1 = Carrier L
and
A6:
v1 = Sum (L (#) F1)
; ( for F being FinSequence of the carrier of V holds
( not F is one-to-one or not rng F = Carrier L or not v2 = Sum (L (#) F) ) or v1 = v2 )
given F2 being FinSequence of the carrier of V such that A7:
F2 is one-to-one
and
A8:
rng F2 = Carrier L
and
A9:
v2 = Sum (L (#) F2)
; v1 = v2
defpred S1[ set , set ] means {$2} = F1 " {(F2 . $1)};
A10:
len F1 = len F2
by A4, A5, A7, A8, FINSEQ_1:65;
A11:
dom F1 = Seg (len F1)
by FINSEQ_1:def 3;
A12:
dom F2 = Seg (len F2)
by FINSEQ_1:def 3;
A13:
for x being set st x in dom F1 holds
ex y being set st
( y in dom F1 & S1[x,y] )
proof
let x be
set ;
( x in dom F1 implies ex y being set st
( y in dom F1 & S1[x,y] ) )
assume
x in dom F1
;
ex y being set st
( y in dom F1 & S1[x,y] )
then
F2 . x in rng F1
by A5, A8, A10, A11, A12, FUNCT_1:def 5;
then consider y being
set such that A14:
F1 " {(F2 . x)} = {y}
by A4, FUNCT_1:144;
take
y
;
( y in dom F1 & S1[x,y] )
y in F1 " {(F2 . x)}
by A14, TARSKI:def 1;
hence
y in dom F1
by FUNCT_1:def 13;
S1[x,y]
thus
S1[
x,
y]
by A14;
verum
end;
consider f being Function of (dom F1),(dom F1) such that
A15:
for x being set st x in dom F1 holds
S1[x,f . x]
from FUNCT_2:sch 1(A13);
A16:
rng f = dom F1
proof
thus
rng f c= dom F1
by RELAT_1:def 19;
XBOOLE_0:def 10 dom F1 c= rng f
let y be
set ;
TARSKI:def 3 ( not y in dom F1 or y in rng f )
assume A17:
y in dom F1
;
y in rng f
then
F1 . y in rng F2
by A5, A8, FUNCT_1:def 5;
then consider x being
set such that A18:
x in dom F2
and A19:
F2 . x = F1 . y
by FUNCT_1:def 5;
F1 " {(F2 . x)} = F1 " (Im (F1,y))
by A17, A19, FUNCT_1:117;
then
F1 " {(F2 . x)} c= {y}
by A4, FUNCT_1:152;
then
{(f . x)} c= {y}
by A10, A11, A12, A15, A18;
then A20:
f . x = y
by ZFMISC_1:24;
x in dom f
by A10, A11, A12, A18, FUNCT_2:def 1;
hence
y in rng f
by A20, FUNCT_1:def 5;
verum
end;
set G1 = L (#) F1;
A21:
len (L (#) F1) = len F1
by Def8;
A22:
( dom F1 = {} implies dom F1 = {} )
;
A23:
f is one-to-one
proof
let y1 be
set ;
FUNCT_1:def 8 for b1 being set holds
( not y1 in proj1 f or not b1 in proj1 f or not f . y1 = f . b1 or y1 = b1 )let y2 be
set ;
( not y1 in proj1 f or not y2 in proj1 f or not f . y1 = f . y2 or y1 = y2 )
assume that A24:
y1 in dom f
and A25:
y2 in dom f
and A26:
f . y1 = f . y2
;
y1 = y2
A27:
y2 in dom F1
by A22, A25, FUNCT_2:def 1;
then A28:
F1 " {(F2 . y2)} = {(f . y2)}
by A15;
A29:
y1 in dom F1
by A22, A24, FUNCT_2:def 1;
then
F2 . y1 in rng F1
by A5, A8, A10, A11, A12, FUNCT_1:def 5;
then A30:
{(F2 . y1)} c= rng F1
by ZFMISC_1:37;
F2 . y2 in rng F1
by A5, A8, A10, A11, A12, A27, FUNCT_1:def 5;
then A31:
{(F2 . y2)} c= rng F1
by ZFMISC_1:37;
F1 " {(F2 . y1)} = {(f . y1)}
by A15, A29;
then
{(F2 . y1)} = {(F2 . y2)}
by A26, A28, A30, A31, FUNCT_1:161;
then A32:
F2 . y1 = F2 . y2
by ZFMISC_1:6;
(
y1 in dom F2 &
y2 in dom F2 )
by A10, A11, A12, A22, A24, A25, FUNCT_2:def 1;
hence
y1 = y2
by A7, A32, FUNCT_1:def 8;
verum
end;
set G2 = L (#) F2;
A33:
dom (L (#) F2) = Seg (len (L (#) F2))
by FINSEQ_1:def 3;
reconsider f = f as Permutation of (dom F1) by A16, A23, FUNCT_2:83;
( dom F1 = Seg (len F1) & dom (L (#) F1) = Seg (len (L (#) F1)) )
by FINSEQ_1:def 3;
then reconsider f = f as Permutation of (dom (L (#) F1)) by A21;
A34:
len (L (#) F2) = len F2
by Def8;
A35:
dom (L (#) F1) = Seg (len (L (#) F1))
by FINSEQ_1:def 3;
now let i be
Element of
NAT ;
( i in dom (L (#) F2) implies (L (#) F2) . i = (L (#) F1) . (f . i) )assume A36:
i in dom (L (#) F2)
;
(L (#) F2) . i = (L (#) F1) . (f . i)then A37:
(
(L (#) F2) . i = (L . (F2 /. i)) * (F2 /. i) &
F2 . i = F2 /. i )
by A34, A12, A33, Def8, PARTFUN1:def 8;
i in dom F2
by A34, A36, FINSEQ_3:31;
then reconsider u =
F2 . i as
Element of
V by FINSEQ_2:13;
i in dom f
by A10, A21, A34, A35, A33, A36, FUNCT_2:def 1;
then A38:
f . i in dom F1
by A16, FUNCT_1:def 5;
then reconsider m =
f . i as
Element of
NAT by A11;
reconsider v =
F1 . m as
Element of
V by A38, FINSEQ_2:13;
{(F1 . (f . i))} =
Im (
F1,
(f . i))
by A38, FUNCT_1:117
.=
F1 .: (F1 " {(F2 . i)})
by A10, A34, A11, A33, A15, A36
;
then A39:
u = v
by FUNCT_1:145, ZFMISC_1:24;
F1 . m = F1 /. m
by A38, PARTFUN1:def 8;
hence
(L (#) F2) . i = (L (#) F1) . (f . i)
by A21, A11, A35, A38, A39, A37, Def8;
verum end;
hence
v1 = v2
by A1, A4, A5, A6, A7, A8, A9, A21, A34, FINSEQ_1:65, RLVECT_2:8; verum