let R be Ring; :: thesis: for V being RightMod of R
for a being Scalar of R
for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Vector of V st k in dom F & v = G . k holds
F . k = v * a ) holds
Sum F = (Sum G) * a
let V be RightMod of R; :: thesis: for a being Scalar of R
for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Vector of V st k in dom F & v = G . k holds
F . k = v * a ) holds
Sum F = (Sum G) * a
let a be Scalar of R; :: thesis: for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Vector of V st k in dom F & v = G . k holds
F . k = v * a ) holds
Sum F = (Sum G) * a
let F, G be FinSequence of V; :: thesis: ( len F = len G & ( for k being Nat
for v being Vector of V st k in dom F & v = G . k holds
F . k = v * a ) implies Sum F = (Sum G) * a )
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 Vector of V st k in dom H & v = I . k holds
H . k = v * a ) holds
Sum H = (Sum I) * a;
A1: S₁[ 0 ]

proof

now

let H, I be FinSequence of V; :: thesis: ( len H = len I & len H = 0 & ( for k being Nat
for v being Vector of V st k in dom H & v = I . k holds
H . k = v * a ) implies Sum H = (Sum I) * a )
assume that
A2: ( len H = len I & len H = 0 ) and
for k being Nat
for v being Vector of V st k in dom H & v = I . k holds
H . k = v * a ; :: thesis: Sum H = (Sum I) * a
( H = <*> the carrier of V & I = <*> the carrier of V ) by A2;
then ( Sum H = 0. V & Sum I = 0. V ) by RLVECT_1:60;
hence Sum H = (Sum I) * a by VECTSP_2:90; :: thesis: verum

end;

hence S₁[ 0 ] ; :: thesis: verum

end;

A3: for i being Element of NAT st S₁[i] holds
S₁[i + 1]

proof

now

let n be Element of NAT ; :: thesis: ( ( for H, I being FinSequence of V st len H = len I & len H = n & ( for k being Nat
for v being Vector of V st k in dom H & v = I . k holds
H . k = v * a ) holds
Sum H = (Sum I) * a ) implies for H, I being FinSequence of V st len H = len I & len H = n + 1 & ( for k being Nat
for v being Vector of V st k in dom H & v = I . k holds
H . k = v * a ) holds
Sum H = (Sum I) * a )
assume A4: for H, I being FinSequence of V st len H = len I & len H = n & ( for k being Nat
for v being Vector of V st k in dom H & v = I . k holds
H . k = v * a ) holds
Sum H = (Sum I) * a ; :: thesis: for H, I being FinSequence of V st len H = len I & len H = n + 1 & ( for k being Nat
for v being Vector of V st k in dom H & v = I . k holds
H . k = v * a ) holds
Sum H = (Sum I) * a
let H, I be FinSequence of V; :: thesis: ( len H = len I & len H = n + 1 & ( for k being Nat
for v being Vector of V st k in dom H & v = I . k holds
H . k = v * a ) implies Sum H = (Sum I) * a )
assume that
A5: len H = len I and
A6: len H = n + 1 and
A7: for k being Nat
for v being Vector of V st k in dom H & v = I . k holds
H . k = v * a ; :: thesis: Sum H = (Sum I) * a
reconsider p = H | (Seg n), q = I | (Seg n) as FinSequence of V by FINSEQ_1:23;
A8: n <= n + 1 by NAT_1:12;
then A9: ( len p = n & len q = n ) by A5, A6, FINSEQ_1:21;

A10: now

let k be Nat; :: thesis: for v being Vector of V st k in dom p & v = q . k holds
p . k = v * a
let v be Vector of V; :: thesis: ( k in dom p & v = q . k implies p . k = v * a )
assume that
A11: k in dom p and
A12: v = q . k ; :: thesis: p . k = v * a
len p <= len H by A6, A8, FINSEQ_1:21;
then A13: dom p c= dom H by FINSEQ_3:32;
dom p = dom q by A9, FINSEQ_3:31;
then I . k = q . k by A11, FUNCT_1:70;
then H . k = v * a by A7, A11, A12, A13;
hence p . k = v * a by A11, FUNCT_1:70; :: thesis: verum

end;

A14: n + 1 in Seg (n + 1) by FINSEQ_1:6;
then ( n + 1 in dom H & n + 1 in dom I ) by A5, A6, FINSEQ_1:def 3;
then reconsider v1 = H . (n + 1), v2 = I . (n + 1) as Vector of V by FINSEQ_2:13;
n + 1 in dom H by A6, A14, FINSEQ_1:def 3;
then A15: v1 = v2 * a by A7;
thus Sum H = (Sum p) + v1 by A6, A9, Lm1
.= ((Sum q) * a) + (v2 * a) by A4, A9, A10, A15
.= ((Sum q) + v2) * a by VECTSP_2:def 23
.= (Sum I) * a by A5, A6, A9, Lm1 ; :: thesis: verum

end;

hence for i being Element of NAT st S₁[i] holds
S₁[i + 1] ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A1, A3);
hence ( len F = len G & ( for k being Nat
for v being Vector of V st k in dom F & v = G . k holds
F . k = v * a ) implies Sum F = (Sum G) * a ) ; :: thesis: verum