let K be Ring; :: thesis: for V being LeftMod of K
for L being Function of the carrier of V, the carrier of K
for A being Subset of V
for F, F1 being FinSequence of the carrier of V st F is one-to-one & rng F = A & F1 is one-to-one & rng F1 = A holds
Sum (L (#) F) = Sum (L (#) F1)

let V be LeftMod of K; :: thesis: for L being Function of the carrier of V, the carrier of K
for A being Subset of V
for F, F1 being FinSequence of the carrier of V st F is one-to-one & rng F = A & F1 is one-to-one & rng F1 = A holds
Sum (L (#) F) = Sum (L (#) F1)

let L be Function of the carrier of V, the carrier of K; :: thesis: for A being Subset of V
for F, F1 being FinSequence of the carrier of V st F is one-to-one & rng F = A & F1 is one-to-one & rng F1 = A holds
Sum (L (#) F) = Sum (L (#) F1)

let A be Subset of V; :: thesis: for F, F1 being FinSequence of the carrier of V st F is one-to-one & rng F = A & F1 is one-to-one & rng F1 = A holds
Sum (L (#) F) = Sum (L (#) F1)

let F, F1 be FinSequence of the carrier of V; :: thesis: ( F is one-to-one & rng F = A & F1 is one-to-one & rng F1 = A implies Sum (L (#) F) = Sum (L (#) F1) )
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: Sum (L (#) F) = Sum (L (#) F1)
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 ;
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] )
proof
let x be object ; :: thesis: ( x in dom F implies ex y being object st
( y in dom F & S1[x,y] ) )

assume x in dom F ; :: thesis: ex y being object st
( y in dom F & S1[x,y] )

then F1 . x in rng F by ;
then consider y being object such that
A14: F " {(F1 . x)} = {y} by ;
take y ; :: thesis: ( y in dom F & S1[x,y] )
y in F " {(F1 . x)} by ;
hence y in dom F by FUNCT_1:def 7; :: thesis: S1[x,y]
thus S1[x,y] by A14; :: thesis: verum
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 A16: rng f = dom F
proof
thus rng f c= dom F ; :: according to XBOOLE_0:def 10 :: thesis: dom F c= rng f
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in dom F or y in rng f )
assume A17: y in dom F ; :: thesis: y in rng f
then F . y in rng F1 by ;
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 ;
then F " {(F1 . x)} c= {y} by ;
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 ;
hence y in rng f by ; :: thesis: verum
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
proof
let y1, y2 be object ; :: according to FUNCT_1:def 4 :: thesis: ( not y1 in dom f or not y2 in dom f or not f . y1 = f . y2 or y1 = y2 )
assume that
A23: y1 in dom f and
A24: y2 in dom f and
A25: f . y1 = f . y2 ; :: thesis: y1 = y2
A28: F " {(F1 . y2)} = {(f . y2)} by ;
F1 . y1 in rng F by ;
then A30: {(F1 . y1)} c= rng F by ZFMISC_1:31;
F1 . y2 in rng F by ;
then A31: {(F1 . y2)} c= rng F by ZFMISC_1:31;
F " {(F1 . y1)} = {(f . y1)} by ;
then {(F1 . y1)} = {(F1 . y2)} by ;
hence y1 = y2 by ; :: thesis: verum
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 ;
( 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;
now :: thesis: for i being Nat st i in dom (L (#) F1) holds
(L (#) F1) . i = G1 . (f . i)
let i be Nat; :: thesis: ( i in dom (L (#) F1) implies (L (#) F1) . i = G1 . (f . i) )
assume A36: i in dom (L (#) F1) ; :: thesis: (L (#) F1) . i = G1 . (f . i)
then A37: ( (L (#) F1) . i = (L . (F1 /. i)) * (F1 /. i) & F1 . i = F1 /. i ) by ;
i in dom F1 by ;
then reconsider u = F1 . i as Element of V by FINSEQ_2:11;
i in dom f by ;
then A38: f . i in dom F by ;
then reconsider m = f . i as Element of NAT ;
reconsider v = F . m as Element of V by ;
{(F . (f . i))} = Im (F,(f . i)) by
.= F .: (F " {(F1 . i)}) by A10, A34, A11, A33, A15, A36 ;
then A39: u = v by ;
F . m = F /. m by ;
hence (L (#) F1) . i = G1 . (f . i) by ; :: thesis: verum
end;
hence Sum (L (#) F) = Sum (L (#) F1) by ; :: thesis: verum