let a, b be non empty XFinSequence of REAL ; :: thesis:
let m be Nat; :: thesis:
assume that
A1: b . 0 is Nat and
A2: len a = m and
A3: len b = m and
A4: a . 0 = b . 0 and
A5: b . 0 < m ; :: thesis:
reconsider k = b . 0 as Nat by A1;
reconsider Fb = XFS2FS* b as FinSequence of REAL ;
reconsider Fa = XFS2FS* a as FinSequence of REAL ;
reconsider c2 = Fa + Fb as FinSequence of REAL ;
A6: b . 0 in Segm m by A1, A5, NAT_1:44;
then A7: len (XFS2FS* a) = b . 0 by A2, A4, AFINSQ_1:def 11;
A8: len (XFS2FS* b) = b . 0 by A3, A6, AFINSQ_1:def 11;
then A9: len c2 = len (XFS2FS* a) by A7, RVSUM_1:115;
then dom c2 = Seg (len (XFS2FS* b)) by A8, A7, FINSEQ_1:def 3;
then dom ((XFS2FS* a) + (XFS2FS* b)) = dom (XFS2FS* b) by FINSEQ_1:def 3;
then Seg (len c2) = dom (XFS2FS* b) by FINSEQ_1:def 3;
then A10: len c2 = k by A8, FINSEQ_1:def 3;
then consider p being XFinSequence of REAL such that
A11: len p = m and
A12: p is_an_xrep_of c2 by A5, Th2, NUMBERS:19;
reconsider b0 = b . 0 as Element of REAL by XREAL_0:def 1;
reconsider p2 = Replace (p,0,b0) as XFinSequence of REAL ;

A13: now :: thesis:

assume k <> 0 ; :: thesis:
thus for i being Nat st 1 <= i & i <= k holds
p2 . i = (a . i) + (b . i) :: thesis:

proof

let i be Nat; :: thesis:
assume that
A14: 1 <= i and
A15: i <= k ; :: thesis:
i in Seg (len c2) by A7, A9, A14, A15, FINSEQ_1:1;
then A16: i in dom c2 by FINSEQ_1:def 3;
( (XFS2FS* a) . i = a . i & (XFS2FS* b) . i = b . i ) by A2, A3, A4, A6, A14, A15, AFINSQ_1:def 11;
then A17: ((XFS2FS* a) + (XFS2FS* b)) . i = (a . i) + (b . i) by A16, VALUED_1:def 1;
( i in NAT & p . i = c2 . i ) by A10, A12, A14, A15, ORDINAL1:def 12;
hence p2 . i = (a . i) + (b . i) by A14, A17, AFINSQ_1:44; :: thesis:

end;

end;

( len p = len p2 & p2 . 0 = b . 0 ) by A1, A5, A11, AFINSQ_1:44;
then m vector_add_prg p2,a,b by A2, A3, A11, A13;
hence ex c being XFinSequence of REAL st m vector_add_prg c,a,b ; :: thesis:
A18: 0 < len b ;
thus for c being non empty XFinSequence of REAL st m vector_add_prg c,a,b holds
XFS2FS* c = (XFS2FS* a) + (XFS2FS* b) :: thesis:

proof

let c be non empty XFinSequence of REAL ; :: thesis:
assume A19: m vector_add_prg c,a,b ; :: thesis:
then consider n being Integer such that
A20: c . 0 = b . 0 and
A21: n = b . 0 and
A22: ( n <> 0 implies for i being Nat st 1 <= i & i <= n holds
c . i = (a . i) + (b . i) ) ;
A23: ( len c = m & ex n being Integer st
( c . 0 = b . 0 & n = b . 0 & ( n <> 0 implies for i being Nat st 1 <= i & i <= n holds
c . i = (a . i) + (b . i) ) ) ) by A19;
then A24: len (XFS2FS* c) = c . 0 by A6, AFINSQ_1:def 11;

now :: thesis:

per cases ;

case n = 0 ; :: thesis:

then ( XFS2FS* b = <*> REAL & XFS2FS* c = <*> REAL ) by A18, A23, A21, AFINSQ_1:64;
hence XFS2FS* c = (XFS2FS* a) + (XFS2FS* b) by RVSUM_1:12; :: thesis:

end;

case n <> 0 ; :: thesis:

set p3 = XFS2FS* c;
for k3 being Nat st 1 <= k3 & k3 <= len (XFS2FS* c) holds
(XFS2FS* c) . k3 = c2 . k3

proof

let k3 be Nat; :: thesis:
assume that
A25: 1 <= k3 and
A26: k3 <= len (XFS2FS* c) ; :: thesis:
A27: c . 0 in len c by A1, A5, A23, AFINSQ_1:86;
then A28: k3 <= n by A20, A21, A26, AFINSQ_1:def 11, A1;
then A29: ( a . k3 = (XFS2FS* a) . k3 & b . k3 = (XFS2FS* b) . k3 ) by A2, A3, A4, A6, A21, A25, AFINSQ_1:def 11;
k3 in Seg (len c2) by A10, A21, A25, A28, FINSEQ_1:1;
then A30: k3 in dom c2 by FINSEQ_1:def 3;
len (XFS2FS* c) = n by A20, A21, A27, AFINSQ_1:def 11, A1;
then (XFS2FS* c) . k3 = c . k3 by A20, A21, A25, A26, A27, AFINSQ_1:def 11
.= (a . k3) + (b . k3) by A22, A25, A28 ;
hence (XFS2FS* c) . k3 = c2 . k3 by A30, A29, VALUED_1:def 1; :: thesis:

end;

hence XFS2FS* c = (XFS2FS* a) + (XFS2FS* b) by A10, A20, A24, FINSEQ_1:14; :: thesis:

end;

end;

end;

hence XFS2FS* c = (XFS2FS* a) + (XFS2FS* b) ; :: thesis:

end;