let W be Z_Module; :: thesis: for V being free finite-rank Z_Module
for A being Subset of V
for B being Basis of V
for T being linear-transformation of V,W
for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds
T . (Sum l) = Sum (T @* l)

let V be free finite-rank Z_Module; :: thesis: for A being Subset of V
for B being Basis of V
for T being linear-transformation of V,W
for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds
T . (Sum l) = Sum (T @* l)

let A be Subset of V; :: thesis: for B being Basis of V
for T being linear-transformation of V,W
for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds
T . (Sum l) = Sum (T @* l)

let B be Basis of V; :: thesis: for T being linear-transformation of V,W
for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds
T . (Sum l) = Sum (T @* l)

let T be linear-transformation of V,W; :: thesis: for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds
T . (Sum l) = Sum (T @* l)

let l be Linear_Combination of B \ A; :: thesis: ( A is Basis of (ker T) & A c= B implies T . (Sum l) = Sum (T @* l) )
assume ( A is Basis of (ker T) & A c= B ) ; :: thesis: T . (Sum l) = Sum (T @* l)
then A1: T | (B \ A) is one-to-one by Th22;
A2: (T | (B \ A)) | () = T | () by ;
then A3: T | () is one-to-one by ;
consider G being FinSequence of V such that
A4: G is one-to-one and
A5: rng G = Carrier l and
A6: Sum l = Sum (l (#) G) by VECTSP_6:def 6;
set H = T * G;
reconsider H = T * G as FinSequence of W ;
A7: rng H = T .: () by
.= Carrier (T @* l) by ;
dom T = [#] V by RANKNULL:7;
then H is one-to-one by ;
then A8: Sum (T @* l) = Sum ((T @* l) (#) H) by ;
T * (l (#) G) = (T @* l) (#) H by A3, A5, Th38;
hence T . (Sum l) = Sum (T @* l) by ; :: thesis: verum