let V be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; :: thesis: for F, G, H being FinSequence of the carrier 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 the carrier 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( set ) -> 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 FINSEQ_1:sch 2();
dom I = Seg (len G) by A4, FINSEQ_1:def 3;
then A6: ( dom G = Seg (len G) & ( for k being Nat st k in Seg (len G) holds
I . k = H₁(k) ) ) by A5, FINSEQ_1:def 3;
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
A7: y in dom I and
A8: I . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A7;
x = - (G /. y) by A5, A7, A8;
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 the carrier of V by FINSEQ_1:def 4;

A9: 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) )
assume A10: k in dom F ; :: thesis: H . k = (F /. k) + (I /. k)
A11: ( dom F = Seg (len F) & dom I = Seg (len I) ) by FINSEQ_1:def 3;
then A12: I . k = I /. k by A1, A4, A10, PARTFUN1:def 6;
thus H . k = (F /. k) - (G /. k) by A3, A10
.= (F /. k) + (I /. k) by A1, A4, A5, A11, A10, A12 ; :: thesis: verum

end;

Sum I = - (Sum G) by A4, A6, Th4;
hence Sum H = (Sum F) - (Sum G) by A1, A2, A4, A9, Th2; :: thesis: verum