let R be non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr ; :: thesis: for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Element of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)

let a be Element of R; :: thesis: for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Element of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)

let V be non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R; :: thesis: for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Element of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)

let F, G be FinSequence of V; :: thesis: ( len F = len G & ( for k being Nat
for v being Element of V st k in dom F & v = G . k holds
F . k = a * v ) implies Sum F = a * (Sum G) )

defpred S1[ Nat] means for H, I being FinSequence of V st len H = len I & len H = \$1 & ( for k being Nat
for v being Element of V st k in dom H & v = I . k holds
H . k = a * v ) holds
Sum H = a * (Sum I);
A1: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; :: thesis: ( S1[n] implies S1[n + 1] )
assume A2: for H, I being FinSequence of V st len H = len I & len H = n & ( for k being Nat
for v being Element of V st k in dom H & v = I . k holds
H . k = a * v ) holds
Sum H = a * (Sum I) ; :: thesis: S1[n + 1]
let H, I be FinSequence of V; :: thesis: ( len H = len I & len H = n + 1 & ( for k being Nat
for v being Element of V st k in dom H & v = I . k holds
H . k = a * v ) implies Sum H = a * (Sum I) )

assume that
A3: len H = len I and
A4: len H = n + 1 and
A5: for k being Nat
for v being Element of V st k in dom H & v = I . k holds
H . k = a * v ; :: thesis: Sum H = a * (Sum I)
reconsider p = H | (Seg n), q = I | (Seg n) as FinSequence of V by FINSEQ_1:18;
A6: n <= n + 1 by NAT_1:12;
then A7: q = I | (dom q) by ;
A8: len p = n by ;
A9: len q = n by ;
A10: now :: thesis: for k being Nat
for v being Element of V st k in dom p & v = q . k holds
p . k = a * v
A11: dom p c= dom H by ;
let k be Nat; :: thesis: for v being Element of V st k in dom p & v = q . k holds
p . k = a * v

let v be Element of V; :: thesis: ( k in dom p & v = q . k implies p . k = a * v )
assume that
A12: k in dom p and
A13: v = q . k ; :: thesis: p . k = a * v
dom q = dom p by ;
then I . k = q . k by ;
then H . k = a * v by A5, A12, A13, A11;
hence p . k = a * v by ; :: thesis: verum
end;
n + 1 in Seg (n + 1) by FINSEQ_1:4;
then A14: n + 1 in dom H by ;
dom H = dom I by ;
then reconsider v1 = H . (n + 1), v2 = I . (n + 1) as Element of V by ;
A15: v1 = a * v2 by ;
p = H | (dom p) by ;
hence Sum H = (Sum p) + v1 by
.= (a * (Sum q)) + (a * v2) by A2, A8, A9, A10, A15
.= a * ((Sum q) + v2) by VECTSP_1:def 14
.= a * (Sum I) by ;
:: thesis: verum
end;
A16: S1[ 0 ]
proof
let H, I be FinSequence of V; :: thesis: ( len H = len I & len H = 0 & ( for k being Nat
for v being Element of V st k in dom H & v = I . k holds
H . k = a * v ) implies Sum H = a * (Sum I) )

assume that
A17: len H = len I and
A18: len H = 0 and
for k being Nat
for v being Element of V st k in dom H & v = I . k holds
H . k = a * v ; :: thesis: Sum H = a * (Sum I)
H = <*> the carrier of V by A18;
then A19: Sum H = 0. V by RLVECT_1:43;
I = <*> the carrier of V by ;
then Sum I = 0. V by RLVECT_1:43;
hence Sum H = a * (Sum I) by ; :: thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch 2(A16, A1);
hence ( len F = len G & ( for k being Nat
for v being Element of V st k in dom F & v = G . k holds
F . k = a * v ) implies Sum F = a * (Sum G) ) ; :: thesis: verum