reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th14:
  for V,W be VectSp of K
  for T be linear-transformation of V,W
  for A be Subset of V for L be Linear_Combination of A st T|A is one-to-one
  holds T.(Sum L) = Sum (T@L)
proof
  let V,W be VectSp of K;
  let T be linear-transformation of V,W;
  let A be Subset of V;
  let L be Linear_Combination of A;
  consider G being FinSequence of V such that
   A1: G is one-to-one and
   A2: rng G=Carrier L and
   A3: Sum L=Sum(L(#)G) by VECTSP_6:def 6;
  set H=T*G;
  reconsider H as FinSequence of W;
  Carrier L c=A by VECTSP_6:def 4;
  then A4: (T|A) | (Carrier L)=T| (Carrier L) by RELAT_1:74;
  assume A5: T|A is one-to-one;
  then A6: T| (Carrier L) is one-to-one by A4,FUNCT_1:52;
  A7: rng H=T.:(Carrier L) by A2,RELAT_1:127
   .=Carrier(T@L) by A6,RANKNULL:39;
  dom T=[#]V by FUNCT_2:def 1;
  then H is one-to-one by A5,A4,A1,A2,FUNCT_1:52,RANKNULL:1;
  then A8: Sum(T@L)=Sum((T@L)(#)H) by A7,VECTSP_6:def 6;
  T*(L(#)G)=(T@L)(#)H by A6,A2,RANKNULL:38;
  hence thesis by A3,A8,MATRLIN:16;
end;
