let b be non empty XFinSequence of REAL ; :: thesis: for a being Real

for m being Nat st b . 0 is Nat & len b = m & b . 0 < m holds

( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) ) )

let a be Real; :: thesis: for m being Nat st b . 0 is Nat & len b = m & b . 0 < m holds

( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) ) )

let m be Nat; :: thesis: ( b . 0 is Nat & len b = m & b . 0 < m implies ( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) ) ) )

assume that

A1: b . 0 is Nat and

A2: len b = m and

A3: b . 0 < m ; :: thesis: ( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) ) )

reconsider k = b . 0 as Nat by A1;

reconsider c2 = a * (XFS2FS* b) as FinSequence of REAL ;

dom (a * (XFS2FS* b)) = dom (XFS2FS* b) by VALUED_1:def 5;

then A4: Seg (len (a * (XFS2FS* b))) = dom (XFS2FS* b) by FINSEQ_1:def 3;

A5: b . 0 in Segm m by A1, A3, NAT_1:44;

then len (XFS2FS* b) = b . 0 by A2, AFINSQ_1:def 11;

then A6: len c2 = k by A4, FINSEQ_1:def 3;

then consider p being XFinSequence of REAL such that

A7: len p = m and

A8: p is_an_xrep_of c2 by A3, 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 ;

then m scalar_prd_prg p2,a,b by A2, A7, A9;

hence ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b ; :: thesis: for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b)

A13: 0 < len b ;

thus for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) :: thesis: verum

for m being Nat st b . 0 is Nat & len b = m & b . 0 < m holds

( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) ) )

let a be Real; :: thesis: for m being Nat st b . 0 is Nat & len b = m & b . 0 < m holds

( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) ) )

let m be Nat; :: thesis: ( b . 0 is Nat & len b = m & b . 0 < m implies ( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) ) ) )

assume that

A1: b . 0 is Nat and

A2: len b = m and

A3: b . 0 < m ; :: thesis: ( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) ) )

reconsider k = b . 0 as Nat by A1;

reconsider c2 = a * (XFS2FS* b) as FinSequence of REAL ;

dom (a * (XFS2FS* b)) = dom (XFS2FS* b) by VALUED_1:def 5;

then A4: Seg (len (a * (XFS2FS* b))) = dom (XFS2FS* b) by FINSEQ_1:def 3;

A5: b . 0 in Segm m by A1, A3, NAT_1:44;

then len (XFS2FS* b) = b . 0 by A2, AFINSQ_1:def 11;

then A6: len c2 = k by A4, FINSEQ_1:def 3;

then consider p being XFinSequence of REAL such that

A7: len p = m and

A8: p is_an_xrep_of c2 by A3, 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 ;

A9: now :: thesis: ( k <> 0 implies for i being Nat st 1 <= i & i <= k holds

p2 . i = a * (b . i) )

( len p = len p2 & p2 . 0 = b . 0 )
by A1, A3, A7, AFINSQ_1:44;p2 . i = a * (b . i) )

assume
k <> 0
; :: thesis: for i being Nat st 1 <= i & i <= k holds

p2 . i = a * (b . i)

thus for i being Nat st 1 <= i & i <= k holds

p2 . i = a * (b . i) :: thesis: verum

end;p2 . i = a * (b . i)

thus for i being Nat st 1 <= i & i <= k holds

p2 . i = a * (b . i) :: thesis: verum

proof

let i be Nat; :: thesis: ( 1 <= i & i <= k implies p2 . i = a * (b . i) )

assume that

A10: 1 <= i and

A11: i <= k ; :: thesis: p2 . i = a * (b . i)

(XFS2FS* b) . i = b . i by A2, A5, A10, A11, AFINSQ_1:def 11;

then A12: (a * (XFS2FS* b)) . i = a * (b . i) by RVSUM_1:44;

( i in NAT & p . i = c2 . i ) by A6, A8, A10, A11, ORDINAL1:def 12;

hence p2 . i = a * (b . i) by A10, A12, AFINSQ_1:44; :: thesis: verum

end;assume that

A10: 1 <= i and

A11: i <= k ; :: thesis: p2 . i = a * (b . i)

(XFS2FS* b) . i = b . i by A2, A5, A10, A11, AFINSQ_1:def 11;

then A12: (a * (XFS2FS* b)) . i = a * (b . i) by RVSUM_1:44;

( i in NAT & p . i = c2 . i ) by A6, A8, A10, A11, ORDINAL1:def 12;

hence p2 . i = a * (b . i) by A10, A12, AFINSQ_1:44; :: thesis: verum

then m scalar_prd_prg p2,a,b by A2, A7, A9;

hence ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b ; :: thesis: for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b)

A13: 0 < len b ;

thus for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds

XFS2FS* c = a * (XFS2FS* b) :: thesis: verum

proof

let c be non empty XFinSequence of REAL ; :: thesis: ( m scalar_prd_prg c,a,b implies XFS2FS* c = a * (XFS2FS* b) )

assume A14: m scalar_prd_prg c,a,b ; :: thesis: XFS2FS* c = a * (XFS2FS* b)

then consider n being Integer such that

A15: c . 0 = b . 0 and

A16: n = b . 0 and

A17: ( n <> 0 implies for i being Nat st 1 <= i & i <= n holds

c . i = a * (b . i) ) ;

A18: ( 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 * (b . i) ) ) ) by A14;

then A19: len (XFS2FS* c) = c . 0 by A5, AFINSQ_1:def 11;

end;assume A14: m scalar_prd_prg c,a,b ; :: thesis: XFS2FS* c = a * (XFS2FS* b)

then consider n being Integer such that

A15: c . 0 = b . 0 and

A16: n = b . 0 and

A17: ( n <> 0 implies for i being Nat st 1 <= i & i <= n holds

c . i = a * (b . i) ) ;

A18: ( 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 * (b . i) ) ) ) by A14;

then A19: len (XFS2FS* c) = c . 0 by A5, AFINSQ_1:def 11;

now :: thesis: ( ( n = 0 & XFS2FS* c = a * (XFS2FS* b) ) or ( n <> 0 & XFS2FS* c = a * (XFS2FS* b) ) )end;

hence
XFS2FS* c = a * (XFS2FS* b)
; :: thesis: verumper cases
( n = 0 or n <> 0 )
;

end;

case
n = 0
; :: thesis: XFS2FS* c = a * (XFS2FS* b)

then
( XFS2FS* b = <*> REAL & XFS2FS* c = <*> REAL )
by A13, A18, A16, AFINSQ_1:64;

hence XFS2FS* c = a * (XFS2FS* b) by RVSUM_1:46; :: thesis: verum

end;hence XFS2FS* c = a * (XFS2FS* b) by RVSUM_1:46; :: thesis: verum

case
n <> 0
; :: thesis: XFS2FS* c = a * (XFS2FS* b)

set p3 = XFS2FS* c;

for k3 being Nat st 1 <= k3 & k3 <= len (XFS2FS* c) holds

(XFS2FS* c) . k3 = c2 . k3

end;for k3 being Nat st 1 <= k3 & k3 <= len (XFS2FS* c) holds

(XFS2FS* c) . k3 = c2 . k3

proof

hence
XFS2FS* c = a * (XFS2FS* b)
by A6, A15, A19, FINSEQ_1:14; :: thesis: verum
let k3 be Nat; :: thesis: ( 1 <= k3 & k3 <= len (XFS2FS* c) implies (XFS2FS* c) . k3 = c2 . k3 )

assume that

A20: 1 <= k3 and

A21: k3 <= len (XFS2FS* c) ; :: thesis: (XFS2FS* c) . k3 = c2 . k3

A22: c . 0 in len c by A1, A3, A18, AFINSQ_1:86;

then A23: k3 <= n by A15, A16, A21, AFINSQ_1:def 11, A1;

then A24: b . k3 = (XFS2FS* b) . k3 by A2, A5, A16, A20, AFINSQ_1:def 11;

len (XFS2FS* c) = n by A15, A16, A22, AFINSQ_1:def 11, A1;

then (XFS2FS* c) . k3 = c . k3 by A15, A16, A20, A21, A22, AFINSQ_1:def 11

.= a * (b . k3) by A17, A20, A23 ;

hence (XFS2FS* c) . k3 = c2 . k3 by A24, RVSUM_1:44; :: thesis: verum

end;assume that

A20: 1 <= k3 and

A21: k3 <= len (XFS2FS* c) ; :: thesis: (XFS2FS* c) . k3 = c2 . k3

A22: c . 0 in len c by A1, A3, A18, AFINSQ_1:86;

then A23: k3 <= n by A15, A16, A21, AFINSQ_1:def 11, A1;

then A24: b . k3 = (XFS2FS* b) . k3 by A2, A5, A16, A20, AFINSQ_1:def 11;

len (XFS2FS* c) = n by A15, A16, A22, AFINSQ_1:def 11, A1;

then (XFS2FS* c) . k3 = c . k3 by A15, A16, A20, A21, A22, AFINSQ_1:def 11

.= a * (b . k3) by A17, A20, A23 ;

hence (XFS2FS* c) . k3 = c2 . k3 by A24, RVSUM_1:44; :: thesis: verum