let V be RealLinearSpace; :: thesis: for a being Real

for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds

G . k = a * (F /. k) ) holds

Sum G = a * (Sum F)

let a be Real; :: thesis: for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds

G . k = a * (F /. k) ) holds

Sum G = a * (Sum F)

let F, G be FinSequence of V; :: thesis: ( len F = len G & ( for k being Nat st k in dom F holds

G . k = a * (F /. k) ) implies Sum G = a * (Sum F) )

assume that

A1: len F = len G and

A2: for k being Nat st k in dom F holds

G . k = a * (F /. k) ; :: thesis: Sum G = a * (Sum F)

A3: ( dom F = Seg (len F) & dom G = Seg (len G) ) by FINSEQ_1:def 3;

for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds

G . k = a * (F /. k) ) holds

Sum G = a * (Sum F)

let a be Real; :: thesis: for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds

G . k = a * (F /. k) ) holds

Sum G = a * (Sum F)

let F, G be FinSequence of V; :: thesis: ( len F = len G & ( for k being Nat st k in dom F holds

G . k = a * (F /. k) ) implies Sum G = a * (Sum F) )

assume that

A1: len F = len G and

A2: for k being Nat st k in dom F holds

G . k = a * (F /. k) ; :: thesis: Sum G = a * (Sum F)

A3: ( dom F = Seg (len F) & dom G = Seg (len G) ) by FINSEQ_1:def 3;

now :: thesis: for k being Nat

for v being VECTOR of V st k in dom G & v = F . k holds

G . k = a * v

hence
Sum G = a * (Sum F)
by A1, RLVECT_1:39; :: thesis: verumfor v being VECTOR of V st k in dom G & v = F . k holds

G . k = a * v

let k be Nat; :: thesis: for v being VECTOR of V st k in dom G & v = F . k holds

G . k = a * v

let v be VECTOR of V; :: thesis: ( k in dom G & v = F . k implies G . k = a * v )

assume that

A4: k in dom G and

A5: v = F . k ; :: thesis: G . k = a * v

v = F /. k by A1, A3, A4, A5, PARTFUN1:def 6;

hence G . k = a * v by A1, A2, A3, A4; :: thesis: verum

end;G . k = a * v

let v be VECTOR of V; :: thesis: ( k in dom G & v = F . k implies G . k = a * v )

assume that

A4: k in dom G and

A5: v = F . k ; :: thesis: G . k = a * v

v = F /. k by A1, A3, A4, A5, PARTFUN1:def 6;

hence G . k = a * v by A1, A2, A3, A4; :: thesis: verum