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 H_{1}( 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 = H_{1}(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 = H_{1}(k) ) )
by A5, FINSEQ_1:def 3;

rng I c= the carrier of V

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

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 H

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 = H

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 = H

rng I c= the carrier of V

proof

then reconsider I = I as FinSequence of the carrier of V by FINSEQ_1:def 4;
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;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

A9: now :: thesis: for k being Nat st k in dom F holds

H . k = (F /. k) + (I /. k)

Sum I = - (Sum G)
by A4, A6, Th4;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;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

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