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 Th42:
  for X being Subset of V, l being Linear_Combination of T .: X, v
  being Element of V st v in X & T|X is one-to-one holds (T#l).v = l.(T.v)
proof
  let X be Subset of V, l be Linear_Combination of T .: X, v be Element of V;
  assume v in X & T|X is one-to-one;
  then ( not v in dom ((V \ X) --> 0.F))& T#l = (l*T) +* ((V \ X) --> 0.F) by
Def6,XBOOLE_0:def 5;
  then
A1: (T#l).v = (l*T).v by FUNCT_4:11;
  dom T = [#]V by Th7;
  hence thesis by A1,FUNCT_1:13;
end;
