let K be Ring; :: thesis:
let V be LeftMod of K; :: thesis:
let L be Function of the carrier of V, the carrier of K; :: thesis:
let A be Subset of V; :: thesis:
let F, F1 be FinSequence of the carrier of V; :: thesis:
assume that
A4: F is one-to-one and
A5: rng F = A and
A7: F1 is one-to-one and
A8: rng F1 = A ; :: thesis:
set v1 = Sum (L (#) F);
set v2 = Sum (L (#) F1);
defpred S₁[ object , object ] means {$2} = F " {(F1 . $1)};
A10: len F = len F1 by A4, A5, A7, A8, FINSEQ_1:48;
A11: dom F = Seg (len F) by FINSEQ_1:def 3;
A12: dom F1 = Seg (len F1) by FINSEQ_1:def 3;
A13: for x being object st x in dom F holds
ex y being object st
( y in dom F & S₁[x,y] )

let x be object ; :: thesis:
assume x in dom F ; :: thesis:
then F1 . x in rng F by A5, A8, A10, A11, A12, FUNCT_1:def 3;
then consider y being object such that
A14: F " {(F1 . x)} = {y} by A4, FUNCT_1:74;
take y ; :: thesis:
y in F " {(F1 . x)} by A14, TARSKI:def 1;
hence y in dom F by FUNCT_1:def 7; :: thesis:
thus S₁[x,y] by A14; :: thesis:

end;

consider f being Function of (dom F),(dom F) such that
A15: for x being object st x in dom F holds
S₁[x,f . x] from FUNCT_2:sch 1(A13);
A16: rng f = dom F

thus rng f c= dom F ; :: according to XBOOLE_0:def 10 :: thesis:
let y be object ; :: according to TARSKI:def 3 :: thesis:
assume A17: y in dom F ; :: thesis:
then F . y in rng F1 by A5, A8, FUNCT_1:def 3;
then consider x being object such that
A18: x in dom F1 and
A19: F1 . x = F . y by FUNCT_1:def 3;
F " {(F1 . x)} = F " (Im (F,y)) by A17, A19, FUNCT_1:59;
then F " {(F1 . x)} c= {y} by A4, FUNCT_1:82;
then {(f . x)} c= {y} by A10, A11, A12, A15, A18;
then A20: f . x = y by ZFMISC_1:18;
x in dom f by A10, A11, A12, A18, FUNCT_2:def 1;
hence y in rng f by A20, FUNCT_1:def 3; :: thesis:

end;

reconsider G1 = L (#) F as FinSequence of V ;
A21: len G1 = len F by VECTSP_6:def 5;
A22: f is one-to-one

let y1, y2 be object ; :: according to FUNCT_1:def 4 :: thesis:
assume that
A23: y1 in dom f and
A24: y2 in dom f and
A25: f . y1 = f . y2 ; :: thesis:
A28: F " {(F1 . y2)} = {(f . y2)} by A15, A24;
F1 . y1 in rng F by A5, A8, A10, A11, A12, A23, FUNCT_1:def 3;
then A30: {(F1 . y1)} c= rng F by ZFMISC_1:31;
F1 . y2 in rng F by A5, A8, A10, A11, A12, A24, FUNCT_1:def 3;
then A31: {(F1 . y2)} c= rng F by ZFMISC_1:31;
F " {(F1 . y1)} = {(f . y1)} by A15, A23;
then {(F1 . y1)} = {(F1 . y2)} by A25, A28, A30, A31, FUNCT_1:91;
hence y1 = y2 by A7, A10, A11, A12, A23, A24, ZFMISC_1:3; :: thesis:

end;

set G2 = L (#) F1;
A33: dom (L (#) F1) = Seg (len (L (#) F1)) by FINSEQ_1:def 3;
reconsider f = f as Permutation of (dom F) by A16, A22, FUNCT_2:57;
( dom F = Seg (len F) & dom G1 = Seg (len G1) ) by FINSEQ_1:def 3;
then reconsider f = f as Permutation of (dom G1) by A21;
A34: len (L (#) F1) = len F1 by VECTSP_6:def 5;
A35: dom G1 = Seg (len G1) by FINSEQ_1:def 3;

let i be Nat; :: thesis:
assume A36: i in dom (L (#) F1) ; :: thesis:
then A37: ( (L (#) F1) . i = (L . (F1 /. i)) * (F1 /. i) & F1 . i = F1 /. i ) by A34, A12, A33, VECTSP_6:def 5, PARTFUN1:def 6;
i in dom F1 by A34, A36, FINSEQ_3:29;
then reconsider u = F1 . i as Element of V by FINSEQ_2:11;
i in dom f by A10, A21, A34, A35, A33, A36, FUNCT_2:def 1;
then A38: f . i in dom F by A16, FUNCT_1:def 3;
then reconsider m = f . i as Element of NAT ;
reconsider v = F . m as Element of V by A38, FINSEQ_2:11;
{(F . (f . i))} = Im (F,(f . i)) by A38, FUNCT_1:59
.= F .: (F " {(F1 . i)}) by A10, A34, A11, A33, A15, A36 ;
then A39: u = v by FUNCT_1:75, ZFMISC_1:18;
F . m = F /. m by A38, PARTFUN1:def 6;
hence (L (#) F1) . i = G1 . (f . i) by A21, A11, A35, A38, A39, A37, VECTSP_6:def 5; :: thesis:

end;

hence Sum (L (#) F) = Sum (L (#) F1) by A4, A5, A7, A8, A21, A34, FINSEQ_1:48, RLVECT_2:6; :: thesis: