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

let V be non empty right_complementable Abelian add-associative right_zeroed ModuleStr over R; :: thesis: for F, G, H being FinSequence of V st len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)

let F, G, H be FinSequence of V; :: thesis: ( len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = (F /. k) - (G /. k) ) implies Sum H = (Sum F) - (Sum G) )

assume that
A1: len F = len G and
A2: len F = len H and
A3: for k being Nat st k in dom F holds
H . k = (F /. k) - (G /. k) ; :: thesis: Sum H = (Sum F) - (Sum G)
deffunc H1( Nat) -> Element of the carrier of V = - (G /. \$1);
consider I being FinSequence such that
A4: len I = len G and
A5: for k being Nat st k in dom I holds
I . k = H1(k) from A6: dom I = Seg (len G) by ;
then A7: for k being Nat st k in Seg (len G) holds
I . k = H1(k) by A5;
rng I c= the carrier of V
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng I or x in the carrier of V )
assume x in rng I ; :: thesis: x in the carrier of V
then consider y being object such that
A8: y in dom I and
A9: I . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A8;
x = - (G /. y) by A5, A8, A9;
then reconsider v = x as Element of V ;
v in V ;
hence x in the carrier of V ; :: thesis: verum
end;
then reconsider I = I as FinSequence of V by FINSEQ_1:def 4;
A10: Seg (len G) = dom G by FINSEQ_1:def 3;
now :: thesis: for k being Nat st k in dom F holds
H . k = (F /. k) + (I /. k)
let k be Nat; :: thesis: ( k in dom F implies H . k = (F /. k) + (I /. k) )
A11: dom F = dom G by ;
assume A12: k in dom F ; :: thesis: H . k = (F /. k) + (I /. k)
then k in dom I by ;
then A13: I . k = I /. k by PARTFUN1:def 6;
thus H . k = (F /. k) - (G /. k) by
.= (F /. k) + (- (G /. k))
.= (F /. k) + (I /. k) by A5, A6, A10, A12, A13, A11 ; :: thesis: verum
end;
then A14: Sum H = (Sum F) + (Sum I) by A1, A2, A4, Th2;
Sum I = - (Sum G) by A4, A7, A10, Th4;
hence Sum H = (Sum F) - (Sum G) by A14; :: thesis: verum