reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;

theorem Th40:
  for A being Subset of V, B being Basis of V, l being
  Linear_Combination of B \ A st A is Basis of ker T & A c= B holds T.(Sum l) =
  Sum (T@l)
proof
  let A be Subset of V, B be Basis of V, l be Linear_Combination of B \ A;
  assume A is Basis of ker T & A c= B;
  then
A1: T|(B \ A) is one-to-one by Th22;
  Carrier l c= B \ A by VECTSP_6:def 4;
  then
A2: (T|(B \ A))|(Carrier l) = T|(Carrier l) by RELAT_1:74;
  then
A3: T|(Carrier l) is one-to-one by A1,FUNCT_1:52;
  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 as FinSequence of W;
A7: rng H = T .: (Carrier l) by A5,RELAT_1:127
    .= Carrier (T@l) by A3,Th39;
  dom T = [#]V by Th7;
  then H is one-to-one by A4,A5,A1,A2,Th1,FUNCT_1:52;
  then
A8: Sum (T@l) = Sum ((T@l) (#) H) by A7,VECTSP_6:def 6;
  T*(l (#) G) = (T@l) (#) H by A5,A3,Th38;
  hence thesis by A6,A8,MATRLIN:16;
end;
