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 Element of 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 Element of 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 & len F = len H ) and
A2: for k being Element of NAT st k in dom F holds
H . k = (F /. k) - (G /. k) ; :: thesis: Sum H = (Sum F) - (Sum G)
A3: dom G = Seg (len G) by FINSEQ_1:def 3;
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();
A6: dom I = Seg (len G) by A4, FINSEQ_1:def 3;
A7: for k being Element of NAT st k in Seg (len G) holds
I . k = H₁(k) by A5, A6;
rng I c= the carrier of V

proof

let x be set ; :: 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 set such that
A8: ( y in dom I & I . y = x ) by FUNCT_1:def 5;
reconsider y = y as Element of NAT by A8, FINSEQ_3:25;
x = - (G /. y) by A5, A8;
then reconsider v = x as Element of V ;
v in V by RLVECT_1:3;
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: Sum I = - (Sum G) by A3, A4, A7, Th6;

now

let k be Element of NAT ; :: thesis: ( k in dom F implies H . k = (F /. k) + (I /. k) )
A10: ( dom F = Seg (len F) & dom I = Seg (len I) ) by FINSEQ_1:def 3;
assume A11: k in dom F ; :: thesis: H . k = (F /. k) + (I /. k)
then A12: I . k = I /. k by A1, A4, A10, PARTFUN1:def 8;
thus H . k = (F /. k) - (G /. k) by A2, A11
.= (F /. k) + (I /. k) by A1, A5, A10, A11, A12, A6 ; :: thesis: verum

end;

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